A tricopter is similar to a helicopter except that it has three vertical rotors as opposed to a heli's one (or two in the case of coaxial). For more information about tricopters, see Bob's (aka signguy) excellent thread  at rcgroups.
The design of a tricopter includes three motors, one of which is vectored.
Here  is a close-up example of a vectored yaw mechanism for a tricopter. Courtesy of rcgroups user bas10an.
The components required to build a tricopter include gyros, ESCs, motors, battery, and a BEC.
The Drive System
The drive system should be configured to provide the highest amount of "thrust resolution" possible. In this context, thrust resolution means the amount of increase or decrease in thrust produced as as the transmitter or gyro instructs the ESC to change power to the motor. A higher resolution (smaller changes) allows the tricopter to make minor adjustments and fly smoother.
The lower the kv rating, the higher the thrust resolution. Motors under 1000kv are ideal for tricopters.
The effect propellors have on thrust resolution comes by virtue of their pitch and weight. Low pitch and low weight propellors provide the highest thrust resolution.
Three ESCs are required, one for each motor. The throttle of each ESC is controlled by a separate channel and has a gyro between the receiver and the ESC. This allows the tricopter to become stabilized by varying the speed of each motor individually.
Some ESCs have finer resolution when instructing the motor to change speeds. There is no documentation for this and has been found through trial and error. Please see the Successful Combos section when selecting ESCs.
A total of four gyros are required, one for each motor and another for vectored thrust on the yaw motor. Unless you use a mixer such a CSM cyclock, in which case 3 gyros are required, aile elev and rudder.
In theory three gyros should be sufficient. Some complicated mixing will then be involved. Because currently, most people configure the mixing to happen inside the transmitter, the gyros will have to handle the pre-mixed signals, therefore requiring an extra gyro. It should not be too difficult to program an AVR microprocessor to take the un-mixed signals, together with a gyro output, to mix and correct the signals. Whether this can be made to work with RC Gyros, or will have to be built on bare-chips-gyros remains to be seen. Here is an example of the bare-chip-gyro: 
While most ESCs include a BEC, only one should be used at a time. To disable the BEC on an ESC, cut the red wire. Some people use a dedicated BEC instead of relying on one from the ESC. Doing so would eliminate the possibility that one of the three ESCs becomes overworked.
One of the functions of an ESC is the monitoring of battery voltage. The typical reaction of the ESC when it senses the voltage is too low is to cut power to the motor. While this might be acceptable in single-motor aircraft, it could cause a tricopter to crash. Due to this, the ESC should be set to NiCad (reference needed), effectively disabling the voltage monior, and a separate battery monitor should be used.
When sizing tricopter batteries, keep in mind that the current demanded by an individual motor will be tripled. Select a battery with 3x the C rating required by a single motor.
Note that Bob only barely follows this rule. He uses a 2200mAh 20C battery able to provide 44A, while his motor/prop will use up to 18A  so three motors would be able to pull up to 54A! (which is over the max rating for the battery, if held for too long).
Tested Component Combinations
In this section, describe your successful electronic component combinations. Please be as specific as possible, including model numbers and retailer links if available.
Bob's proven combo
All the details about Bob/signguy's setup can be found in his build thread. 
- Motor - hexTronik DT750 
- ESC - HobbyKing SuperSimple 25-30amp 
- BEC - TURNIGY 3A UBEC w/ Noise Reduction 
- Motor Gyros - Telebee Head Lock & Standard Gyro 
- Yaw Gyro - GWS PG-03 Piezo Gyro 
- Battery - ZIPPY Flightmax 2200mAh 3S1P 20C
- Battery Monitor - Hobby King Battery Monitor 3S
- Propellors - 11X4.7 GWS slow flier
The HK gyro  works according to some people, others say to avoid it. The HK gyro is now used widely and is the accepted gyro for use with the DT750 and SS25 - 35 ESC's
Some gyro/ESC combinations do not work for tricopters. Please detail your unsuccessful combos.
Note that this is a difficult item: If it didn't work for someone they might have had something connected or configured wrong. So it might still work for someone else. So if you don't get it to work, please report your FULL configuration. Including transmitter, receiver, battery ubec. Even if swapping the Gyro solved all your problems that doesn't mean it's the Gyro. it could still be the gyro / ESC combo for instance, where swapping either would have worked.
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