Yes the ESP8266 will connect to a Cell Phone , I can connect to it through either my Android , iPhone , home wifi router or laptop in either a direct connection or AP mode

I flashed the ESP8266 with LUA using ESPlorer and set it up as a server .

Here is the code I use to send and receive data to/from wifi / usb.

uart.setup(0,9600,8,0,1,0)
sv=net.createServer(net.TCP, 1800)--1800=30min timeout.
global_c = nil
ip=wifi.sta.getip()
print("?")
print(ip)
print("/")
sv:listen(9999, function(c)
if global_c~=nil then
    global_c:close()
global_c=c
c:on("receive",function(sck,pl) print(pl) end)
uart.on("data",0, function(data)
    if global_c~=nil then
        global_c:send(data)
end, 0)
I’ve used the previous .5MB version and now this new 1MB version. I also added a review of this module spanning both versions. My first use of this module was with a 3.3v USB to serial connected directly to the module. As I mentioned in my review it was rock solid.
Next I moved on to controlling it with a PIC micro-controller, I wasn’t having much luck as they seemed to keep rebooting due to watch dog or crash with exception dumps. They would also get very hot and the Red LED would get very dim. I was getting very close to contacting Sparkfun tech support that these modules have an issue, however if I put them back connected directly to the USB to serial they worked fine. At that point I decided it must be something to do with the power and I hooked up the ESP8266 with its own 3.3v regulator and I am happy to report these modules are as solid as when hooked directly to the USB to serial interface. It doesn't matter how much current your 5V source can supply to your regulator, this module seems to be either sensitive to noise and or it uses power in surges for the transmitter and if the voltage momentarily drops the board goes into some sort of latch up condition. If you see resets and the power LED going dim, try a dedicated regulator, you'll be glad you did.
IMPORTANT: If you want to keep your sanity when working with these on a breadboard, supply power to them with their own 3.3v regulator, probably not a bad idea to do the same when making a PCB board either.
Also most people seem to use this module with TCP connections. I am using it with UDP sockets and there isn’t a lot of information out there. If you need information on using UDP sockets check out this series of articles I have published on a prototype I am creating using the ESP8266 with UDP. Also you can check out my Google+ collection on the ESP8266 prototype here.
I've discovered that if you put a ~2k resistor between the CH_PD and power, the board stays significantly cooler. I used to get nervous running mine, so I constantly had a fan blowing on it, but after I read several tutorials recommending the use of a resistor, it's been fine since then. It still gets warm, but not burning. I'm pretty sure the resistor I had on it was a 2k, but I'm sure if you experiment around with different resistances you'll find one that works for you.
Also, make sure you are using 3.3v everywhere, since the board is not 5v tolerant.
-This is all assuming you don't in fact have a defective board. :) Good luck!
          
UPDATE:
I've used the now retired .5MB version and this version. Below I have put a copy of my original review. Since posting that I moved past just controlling it via a USB to Serial interface and put it on a breadboard with a microcontroller and I found this to be important: Use a dedicated 3.3v regulator to power this module, see my comment in the comments section about this.
PAST REVIEW:
A lot of bang for the buck! Overall it’s a pretty solid little board. It comes preloaded with firmware that implements an AT command interface. I recommend updating to the latest AT command firmware, especially if you are going to use UDP mode of communication.
I haven’t used it with TCP connections, but the UDP mode seems to be very robust. I’ve had it running for over 3 days sending UDP packets every 2 seconds and it’s still going strong, no hiccups with WiFi connection or serial data transmission to and from the module. I’m sure my awesome software design and coding has something to do with it, but the module is holding its own :)
Range seems pretty good, I had no problem connecting to my router indoors from anywhere inside, and I even took my setup on a breadboard out to the furthest point in my yard, about 150 feet from the house and was able to connect to my router down in the basement through solid concrete poured foundation. Not too shabby.
Since they are so economical I bought 2, and I’m glad I did, as the description mentions do not hook this up to more than 3.3 volts. When I went to update my firmware on one of the boards I didn’t have my glasses on and inadvertently connected the GPIO0 to USB 5v rather than ground the next terminal over. When I noticed the mix-up I switched it to ground and successfully flashed the new image, but when I try to boot up normal it just spews garbage out of the serial port, so it is very sensitive to the wrong voltage.
I don’t have any issues with the board, but I can’t bring myself to give it a 5 star rating as I believe that should be saved for special circumstances, even though this module is lots of bang for the buck I feel like there could be a better ESP8266 board so I’ll call it a 4.5 and recommend you play around with one if you haven’t already.
          
RX TX pins were reversed for me at least if you go by NURDSpace. I connected the pin labeled RX to TX on my Arduino and TX to RX (all through level shifter) and it didn't work, I switched the pins RX to RX and TX to TX and now it works.
Works great now that I've straightened that out, I was ready to ask for return on a defective module.
          
The community has really come together to make this inexpensive module easy and (relatively) reliable. I like the Arduino port, because it's easy and works with a whole mess of libraries, but NodeMCU is acceptable, as is the native toolchain. Works well with services like Blynk if you want to control it with a smartphone without writing a custom app and server. Definitely get it some power of its own, nearly turned my FTDI Basic to slag just teasing the NodeMCU upload.
          
Look. I like this module. I really do. Even though I totally fried mine after like three days. It was probably completely my fault.
I've been playing with various ESP8266 modules for a while now. This module is probably exactly what you need if you want to interface with an Arduino and chat with the pre-flashed Hayes AT-based firmware to access web pages. You know, get up and running quickly.
That said, unless you're really a fan of the AT-based firmware, I'd recommend you pass this guy up in favor of the Thing Dev board. Did you know you can upload Arduino sketches to ESP8266 modules pretty easily? But putting these guys in flash-mode is a bit of a chore, and the Thing Dev board is supposed to do that automatically (I don't have one so I haven't verified).
Also I'd prefer more GPIO than this ESP-01 module offers. The Thing would probably be better for that, but if you're looking for something more modular I might recommend just going to eBay and finding a reputable dealer for an ESP-12 or ESP-13.
          
I'll hold off on saying what you've probably already read with the words "cheap", "IoT" and "the future" and tell you what I have learned so far.
I ordered one of these in early October (2015) and was able to get it responding, however, I was thrown off by its version response. I can't speak for other suppliers, whose product images often show a more "plain" blue board, but Sparkfun sells this nifty "black AI-Cloud" (AI-Thinker) version (with possibly a larger flash than "blue" versions, which can be important later on for Firmware Over The Air (FOTA) and other firmware flashing). I believe AI-Thinker is partnered/licensed by some means with Espressif (the ESP8266 SoC maker) to make a board. I compare it to Arduino using Atmel's AVR. I mention this because support for this chip is largely community driven and it can become very confusing. You'll see that you can use AT commands initially to talk to the device and the firmware version on these is slightly different than 99% of what you will read; it seems to be AI-Thinker's version of Espressif's AT commands. The commands seem to be the same and I don't know the differences, other than the possibility that these boards get their FOTA from AI-Cloud's servers instead of Espressif's servers.




    
Considering the versions and the fact that these (and similar modules) are constantly being updated, the module I received was based off of

  
AT version 0.25 (Jun 5 2015) SDK version 1.1.1
and FOTA upgraded to

  
AT version 0.30 (Jul 3 205) SDK version (Build) 1.2.
Since I am brilliant I nearly broke my module and decided to then order an additional three at the end of the same month (October). I have so far only tested one of these devices and the version on it is

  
AT version 0.40 (Aug 8 2015) SDK version (Build) 1.3 Sep 11 2015.
Good to know these modules are up to date AND which firmware we are working with. Also important since the community will tell you these modules have varying baud rate settings, the baud on these was set to 115200.
In case you haven't figured it out yet, there are many different firmware versions out there, including third-party. Espressif has their firmware, SDK and flasher, AI-Thinker has their iterations (as I'll just call them) on Espressif's firmware, SDK and flasher. You also have NodeMCU firmware (based on the Lua programming language), micropython (based on Python), ESP Basic (based on the BASIC programming language), Espruino (based on the development board of the same name and the JavaScript language) and of course an Arduino Add-on. Very awesome that there are options, but can be another confusing thing you can come across.
If this is the first time you are working with this board, grab a 3.3V Basic FTDI, DEV-09873 $14.95, 3.3V Regulator, COM-00526 $1.95, a Bi-Directional Logic Level Converter, BOB-12009 $2.95 and some Male/Female Jumper Wires - 20/PK, PRT-12794 $1.95. The extra $22 may save you some peace of mind and can certainly be used for other projects. I did initially use the logic level converter and was able to work off of the Arduino's 3V3 pin, however, some quirks I ran into (which I believe lead to further problems) could probably have been attributed to the Arduino unable to provide the necessary current causing the module to reset. NOTE: This doesn't mean I had a problem with the module, rather the problems were very likely user related.
I did not give this module a 5 star rating because I'm still learning about this module and I'm an electronics hobbyist so my knowledge may not be as great. Therefore, I didn't think it was fair to give it a full rating if I haven't been able to use all of it's features. Yes... It's not breadboard friendly but the M/F jumper wires makes it a non-issue. Otherwise I was able to connect it to the internet out of the box and all for $7!
EDIT: Fix some formatting.
          
I bought two, and I can't program them. It appears that the reset pin has not been connected to the header. Pin 32 is the reset pin on the chip and it goes to what appears to be a 10K pullup resistor and and small cap that goes to ground. But, it does not go to the pin on the header that the documentation describes. Hence, I can't program it. Waiting on tech support. I can communicate with it, but reset seems to be broken.
edit 7/11/2017 Turns out tech support never got my request, but contacted me because they saw my review. I discovered that the reset line has a 12k resistor in between the pin and the micro. There also is a pull up (10K). No way to pull the reset line low, so I removed the 12k resistor and replaced it with a short. I can program the part now. Leaving my review rating as it is because the modules that I received were built wrong. Your mileage may vary. The resistor is the one closest to the header pin 1.
          
I bought a couple of these modules last month (Nov. 2015) and when I finally got around to exploring with these, it took me a few days to get them working.
My "Be Aware" remark in the title is because this is not something that an inexperienced user will find easy to work with.  A few things to point out.
The module is very picky about power.  I recommend powering this with a dedicated power supply.  I also recommend using a dedicated power supply along with filtering capacitors to ensure that the power is steady.  When I finally got the modules to communicate, there were numerous resets.  Good jumper wires and good connections to the breadboard along with filtering capacitors solved the problem.
Documentation is shall I say "not really up to speed".  I have found that there are pictures of the pinouts on the internet that are wrong.
Once you do get one of these working properly... WOW!  The possibilities are endless.
The only reason that I gave this 4 stars rather than 5 is because of the difficulty and frustration involved getting them to work.  I must point out that my purchase from Sparkfun was a good experience as always, and they did have tech support on hand to help me out.
          
It's amazing that this board provides the level of Wi-Fi and TCP/IP functionality for the price.  That said, it is tricky to get started using the device.  I found that my module did not work properly until the CH_PD pin was pulled up to VCC.  The Nurdspace page that is linked from the product page doesn't list this pin.  I also found that I had to flash the device before it worked properly.  I'm not sure if it was unprogrammed when I received it or if I did something to mess it up while trying to get it connected properly.
I found this website extremely useful as a reference for flashing the board and for many other odds and ends:  http://labdegaragem.com/forum/topics/conhecendo-o-esp8266  Note that this board (or equivalent) is named ESP8266-01.  The website is in Portuguese but Google Translate did an excellent job and it's quite readable after being translated.  I achieved first-time success in flashing the board by following the instructions on this page.
One other thing--the page I linked as well as some forum posts note that PuTTY and Tera Term don't agree with the ESP8266.  A free utility, TERMITE (linked from the above page) worked well with it.
          
It's a very good product and I'm having tons of fun with it. However, as mentioned in previous comments, you need to be patient when you try to connect it the first time.
There are all sorts of things that can go wrong and here's what I found until it worked properly.
1) Power Supply: make sure to provide enough juice to the module and to apply a large filtering capacitor between its VDD and GND (1uF worked for me).
2) Pinout: the pins I found in the websites provided were correct except for at least one that was ignored. The image you may find (top view) look like this
(Antenna on Top)
GND_ NC NC _Rx
TX_ NC NC _VCC
But at least one of the "NC" should be connected during setup. It's called CH_PD. This pin has to be pulled-up to VCC to enter AT command mode. You pull it down to GND to be able to change the firmware (didn't try it yet). So the pins now look like this:
(Antenna on Top)
GND_ NC NC _Rx
TX_ CH_PD NC _VCC
3) Baud Rate: the baud rate or this device is verified as described to be 115200. But if you get one from other vendors, the firmware version might be different with a different baud rate.
4) Terminal Program: I used my pc to connect to the device. There seems to be data encoding issues with some terminal programs. Tera Term didn't work for me so I switched to one some people suggested called TERMITE (it's actually cool). I setup everything and finally got a response for my "AT" command. But it was always showing "ERROR". I fixed this after changing the settings of Transmit and Receive in TERMITE to "CP + LP" (the default comes either CP or LP only).
With all this it worked well and I was able to communicate with it and configure its network connection later.
          
I've seen the other reviews and searched the internet along with contacting help at Sparkfun.  However, I could never get them to work.  The module only returns garbled text when sent an AT command.  All the snippets of help said the same thing, that it is either power issue or baud rate.  I have provided its own power supply at 3.3 with 1amp.  I have tried all baud rates.  I even tried to manually change the baud rate by sending the proper AT command to do that and I still could not get anything readable.
The Sparkfun ESP8266 WiFi Module doesn't work correctly, it can't connect to Hostpot (AP WiFi) neither to the STA WiFi, it takes too much time, is not reliable to all.
Personally I usually buy from Digi-Key, not from your website directly,
Digi key Part Number            1568-1235-ND, 
Manufacturer                            SparkFun Electronics
Manufacturer Part Number    WRL-13678
Your Module is low quality compare to 
Digi-Key Part Number                       1597-1145-ND
Manufacturer                                       Seeed Technology Co., Ltd
Manufacturer Part Number               317060016
This is working perfectly.
Why did I buy directly from your website? Is because after wasting 100 pcs of your Modules, I decided to buy from your website maybe the quality can be better than the one you have on Digi Key. This is why I bought 10 pcs for a test directly from you website, your modules are not working at all.
The other manufacturer Module works perfectly without giving any connection problems I bought from Digi Key lot of modules, unfortunately the modules were sold out by the moment I wanted to order.
Best Regards
          
Module came quickly in great condition, I'm having a little trouble getting it online but I blame my news status to arduino and coding not the product. I would definitely recommend it if you are in the market for adding wifi to a project. It can also be set up as a stand alone controller which I find really cool. The main downside is the inability to convert logic signals from 5 to 3.3v onboard so you have to use a logic level converter. Overall a great wifi option!
          
I've never worked with anything over serial yet, but holy smokes was this thing easy to setup. Right now I'm just messing with it through the Serial Monitor via an arduino, and it was able to seamlessly list all networks around. Connect to my wifi. And accept all the commands with zero problems. I'm so impressed with this thing, I think I'm going to buy lots more. The possibilities are endless here!!
          
I am using this module with esp-link to connect to a HY-TinySTM103T running mecrisp-stellaris forth.   I initially had trouble flashing firmware onto the board until I discovered that the flash mode had to be set to 'dio' or 'dout' rather than 'qio' (the default in many cases).   None of the documentation listed on the sparkfun site indicates this, I had to determine it experimentally.
          
So the ESP8266 was a little difficult to get linked up all right. There are tons of conflicting bits of documentation all for different flavors of the ESP8266. I successfully used this guide for the "Thing", but it worked perfectly: https://learn.sparkfun.com/tutorials/esp8266-thing-hookup-guide/installing-the-esp8266-arduino-addon
One of the things that wasn't super clear to me right away was that you don't actually need an arduino processor to communicate between the FTDI and the on-board processor for the ESP8266. You can flash it directly from your FTDI.
I used a simple FTDI bus with a 3V external power supply (to power the wifi module) across a 1000uF capacitor. I used 100ohm resistors on the data lines. I had no trouble damaging the circuit even though the FTDI spits out 5V on the RX/TX pins. I used the FTDI basic with the full-size USB port.
          
This module does exactly what I wanted...connects to the WiFi and sends me an alert when an event happens. Perfect tool for tge job and good price.
The only negative thing I have to say is about the GPIOs. It is not very clear how the GPIOs work, and depending on how you use one pin, the other may not work the same way. Digging into the ESP8266 datasheet will answer these questions, but using the module's GPIOs in the Arduino IDE was not as simple as using a typical Arduino's GPIOs.
          
We love this little WiFi module over here at the geekpower/WhenYouFly studio. In designing a new-fangled replacement for the paper logs many private pilots keep in their airplanes we decided it needed to upload data via WiFi and thus began the search for a small, inexpensive, simple WiFi doodah that would meet our pretty straightforward requirements. We mated it to an Atmega 1284 on our own PCB and simply send it AT commands (remember those from your C64/BBS days?). We run our Atmega at 8Mhz for power reasons, so we keep the data rate between this module pretty slow (9600bps) and it works perfectly. It even has pretty good range - planes outside the hangar area up to about 200' away can upload logged data with good success. Thank you, Spark Fun!
              replied on March 25, 2019:
            
Yes, I remember the C64 and BBS! Dialup at 1200 baud was slow, but got the job done back in those days. :-)
Thank you for the review and for sharing your project!
          
          
This module is absolutely amazing considering the price, but it was a bit tricky to get working. This module required a bit more work than others to get working but it operates perfectly. There is not much documentation online so I would not recommend this module for beginners. Other than the huge learning curve, this module is the best wifi module on a budget.
          
This thing, as mentioned, is "not breadboard friendly".  That is a huge understatement!  Not only is not not breadboard friendly, but it's back extends out too far, so it covers up and potential headers.  I recommend a socket for this.  Don't be like me!
PS: or put it in sideways, and IGNORE the middle pins!
          
...considering it's size and affordability. While it is a little buggy and potentially difficult to program without the right tools, once you get it working, it connects to your wireless network and is running your code within 6 seconds of powerup.
If you need other antennas/controller, there are other models of ESP out there, but this one is perfect for someone seeking a learning opportunity with 2  GPIO pins.
          
Have 2 now. One pulls 20 to 30 ma (no wifi state). other is pulling 93 ma. Gonna get 2 more and hope to figure it out. I'm using pic mcu to send AT commands.. 1st tried Voltage Divider for TX: eventually fried the ESP. Then tried running the PIC at 3.3v: similar result. Now using bidirectional level converter. This  is the ESP pulling 93ma in no wifi state.
          
I am attempting to use this project in conjunction with an Arduino to provide wifi connectivity to one of my projects.  The devices are pretty cheap, which is good.  That said, they can be a bit difficult to get to work.  You will need to be comfortable with "AT" commands, and will probably want to re-flash the device to get the latest firmware.  That said, they are not too bad.  Note that you will probably want a usb to serial cable (FTDI) in order to update the firmware.  Also, the devices are sensitive to voltages.  I'm using a 3V voltage regulator and have not had issues.  Good luck!
          
I like these a lot.  Sure, you could pay a couple extra bucks for the full breakout - and you should if your project requires more GPIO pins.  However, for simple WiFi tasks that requires only a single GPIO (e.g. One-Wire), these are great.  They run cooler and are smaller so they'll fit in smaller places and tighter circuits.  I have many of these around the house as thermometers that report to a server that controls the air conditioner as well as servos (to open / close vents) in order to keep all of the rooms at their desired temperatures.
Keep in mind that while it has a pretty fast clock for an soc, it is extremely picky about watchdog.  So, Serial IO may have you pulling your hair out due to the constant watchdog restarts.  Get past this by not doing both input and output in the same bracket.  I could be wrong, but I think the watchdog is fed every time the stack level is changed.
          
Connects to the network very well after issuing an inordinate number of commands that it should remember.  It also requires termination characters of <cr> <lf> pairs for the commands and for data received and transmitted, both in client and server modes.  There should be a mode that it just transmits and accepts characters as they are sent/received.  Even in "stream" mode a <cr> <lf> pair is required ... absurd.

But if your application doesn't care the module is a great buy and works well.

Don't forget to RS232 interface chip unless going directly to a microprocessor.
          
coming from a background of trying to get the NRF24L01 module to work.  Emphasis on the word "trying".
This is working with an operating system.  Just rewrote it, so still going through bugs here and there.  Need to debug the drivers for this, but they're simple RS-232 drivers for the OS, which should work.  Past that, need to get networking/connect and perhaps RF mesh working.
So far, the chipset looks easier to use, although unless you dig for it, the higher level software is missing.  If you're in an Arduino environment (and I'm not), there may be drivers out there.  Since I'm not, it's roll your own time.  (answer to question:  Yes, the OS has enough goodies that I'll keep it, or perhaps maybe (or not) rewrite it into FreeRTOS, but FreeRTOS doesn't have things I need.)
You'll need RS-232 communication, then something to call functions to connect, then transfer messages.  What you do beyond this is a function of what you decide to do overall.  The creators do have software and infrastructure to look at, but I haven't done that enough yet.
Stay tuned, if I can add to this review....
So far, looks good.  All the work will be up at the system level (message types, connects, dialogues between nodes...)