To clarify, in a reverse flow combustion chamber the fuel and compressed air enter and combust in the opposite direction to the compressor intake. The hot gases then change direction once again to drive the turbines and exit out the back.
As another poster explained, this has the advantage of packing the combustion path into a smaller space (length wise) and also accelerates the hot gases before they reach the turbine stages. The disadvantage is the extreme heat to which the gas flow diverter is subjected.
Rob