http://www.landriderbikes.com
They were very heavily advertised several years ago but currently they seem to show up more on craigslist than on TV.
bicycles.stackexchange.com/users/3083/mikes
I have ridden one - it used weights thrown out by centrifugal force and springs to move the derailleur in and out.
Horrible is all I can say.
Maybe it was me not being used to it, but things like not being in the gear you left it in, and less than smooth changes - which you have not idea when they are going to happen, especially under power.
The only redeeming feature was it was a borrowed bike I could give back at the end of the weekend holiday.
That said, for cruising around a by someone who otherwise would not ride, and only on flat, even ground), its would do the job.
bicycles.stackexchange.com/users/3924/mattnz
Trek had a bicycle a few years ago named "Lime" which had 3 speed automatic gearing.
I don't think it sold well.
It used a gearing system called "Coasting" that was created by Shimano and actually controlled by a computer chip from signals from the front hub.
"A dynamo is fitted on the front hub that gauges the revolutions of the wheel.
It sends this information to a computer chip housed near the pedals on most of the bikes.
From there, the chip, which controls the planetary gears located on the back hub, determines whether to shift up or down.
All the chip needs to make its determination is for a rider to pedal four or five times, according to Shannon Bryant, Coasting project coordinator for Shimano."
bicycles.stackexchange.com/users/2953/Glenn Gervais
There's the Nuvinci Harmony .
It uses the Nuvinci N360 CVP hub, which is a continuously variable transmission , meaning there are no shift points.
The Harmony controller changes the ratio based on cadence, or it can be adjusted manually.
bicycles.stackexchange.com/users/5669/David Holdeman
There have been attempts over the years, but never particularly successful.
One
I recall used a 5-speed rear hub that was shifted by weights on the spokes, similar to a centrifugal governor.
I expect that, with the new electric shifters, there will be some new attempts at it in the next year or two.
With a computer it should be possible to be reasonably "smart", especially if mated to some sort of torque meter.
bicycles.stackexchange.com/users/1584/Daniel R Hicks
I know Im a little late for this posting
but this is the first time I came across such question.
I do own a Trek Lime, purchased back in 2007.
This is a fully automatic bicycle.
It has 3 speeds a front hub dynamo (speed sensor), a shifter module-solenoid (computer) under the center frame, and the 3 speed auto-shifter hub in the rear wheel.
As far as I know there has always been autoshifting bikes out there, they're rare and not very cheap, but a joy to ride.
bicycles.stackexchange.com/users/20836/Raul Huerta
As the Nuvinci system has been mentioned in other answers, I'll mention one more.
SRAM makes the Automatix hub now.
It's a 2 speed system (ratios 1:1, 1:1.37) with a centrifugal clutch.
There's no manual shifting possible and no cables involved.
bicycles.stackexchange.com/users/8219/Batman
Check out the new Autobike http://www.autobike.tv .
It uses a CVT.
bicycles.stackexchange.com/users/8314/user8314
I have one with an automatic, its a six speed.
It works really well actually, you just pedal nice and it does a real nice up/down shift in the gears; its not a hard riding bike, it's a nice causal ride, great for me and the dog, but it will up shift and down shift on its own.
It seems to be an older bike.
I picked it up at a garage sale for ten bucks; it has 3 weights on the back wheel that have the name dad on the and small piece of plastic the size of a dime that rides in a groove.
If I'm not riding it wont shift.
The only stickers on the bike are ones that say metro and automatic.
I have searched the web for about three years and find nothing of the likes of it.
bicycles.stackexchange.com/users/25124/robert
Not really, the one from Nu Vinci is closest but Has manual shift
the weight factor is an issue.
The major problem in developing is the input power is too low for a fully mechanical system to respond smoothly and if you include electronics it becomes too complex.
But I believe some solution is possible.
bicycles.stackexchange.com/users/13866/Sanjoy Nundy
I don't know much about it and it doesn't seem to be publicly available yet
but Bioshift is the most recent variant I know of.
It basically seems to be an extension for Di2 electronic shifting.
http://www.baronbiosys.com/
bicycles.stackexchange.com/users/15828/David Triebe
Has anyone ever designed or manufactured an automatic transmission (or automatic shifter?) for cycling?  I imagine such a device would automatically change the gearing to allow the rider to maintain constant cadence or constant torque on the cranks.  I used to daydream about the existence of such a system many years ago when riding my first (battered, second hand) racer as it was such an ordeal to change gears.
Is there such thing as an automatic transmission for bikes?
bicycles.stackexchange.com
<derailleur><drivetrain><shifting><cadence>
bicycles.stackexchange.com/questions/10381
bicycles.stackexchange.com/users/4537/Ken
There are a number of bikes on the market with automatic gear.  Trek, Shimano, SRAM are some of the companies that make them. Real cycling enthusiasts do not seem to be terrible keen on them. However, for people who would otherwise not cycle, they are a great idea. They are most suited to casual riding, on flat ground but are not really suited to hard riding.  There are several automatic gear systems that use mechanical or electronic shifting. Mechanical options can be difficult to get used to and may change gears suddenly at unexpected times, while electronic shifting can be overly complex. They have never been particularly successful commercially, but can be used with slower bikes for cruising.
There are several automatic gear systems that use mechanical or electronic shifting. They have never been particularly popular, but can be used with slower bikes for cruising. Mechanical options can be difficult to get used to and may change gears suddenly at unexpected times, while electronic shifting can be overly complex. 