Audi of Germany only manufactures water-cooled internal combustion engines for automobiles. As such, any overheating problem with the engine will be related to the coolant system. "Cool" for a running engine is a relative term. What's needed is for the Audi to keep its coolant water about -1.11 degrees C below boiling, or 82.2 degrees Celsius. This allows the water to carry away the engines excess heat and keep it from overheating. The cooling system is composed of four major components: the radiator, water pump, radiator fan and thermostat.
Whenever an Audi begins to overheat or even run hotter than normal, first check the radiator. Make sure the coolant (a mixture of water and chemical coolant/antifreeze) is at the correct level. If the coolant is low, add more to the overflow tank on the side of the radiator. Also, when the motor is not running and when the motor is warm, check under the radiator and see if there is any excess water puddling. It's best to do this without the air conditioner running as normal condensation dripping to the ground could be confusing.
This could indicate a radiator leak. Also, check all the rubber hoses leading from the radiator to the engine block for leaks. Any loss of coolant can result in a hot running engine.
The water pump is located on the side of the engine and is easily located by following the hoses to and from the radiator. The water pump pulls water from the radiator to the engine and then pushes water back to the radiator to be cooled. The water pump operates by a flywheel driven by the serpentine belt. With the engine running, make sure the belt is driving the wheel and the wheel is driving the shaft entering the pump. Also, check for any leaks around the pump. When a water pump fails, the seals keeping it watertight will lose integrity. A leaky water pump must be replaced.
A fan located very close to the radiator face pulls air through the radiator to cool the hot water. Audi engines use electrically-powered fans and not fans driven by the serpentine belt on many older model cars. With the motor running, make sure the fan is operating. If it is not, check the wiring connections powering the fan. If the wiring is functioning, then replacing the fan is the only option.
The thermostat is a small round device located between the main water inlet hose and engine block. The other end of this hose leads to the water pump. Unfortunately, there is no way to test to see if the thermostat has failed. Fortunately, thermostats are very inexpensive and easy to replace. Just disconnect the hose, pull out the old unit and install a new one. This is the best way to troubleshoot the thermostat and it's not much more expensive than replacing an oil filter.