All rules of normal chess apply. However, the board is colored in several colors - like a rainbow.
Besides the normal checkmating there is another winning way: If for one of the colors a player has no piece on a square of this color, he has lost.
 
This rainbow conditions count even higher than the normal 'checkmating': If in the same moment Player A is checkmated and Player B has no piece on say a yellow square then Player B has lost.
Independently of this priority setting most games are decided by a rainbow condition. The drawing quota in "Chess on the Rainbows" is extremely low.
 
After some piece exchanges it really becomes difficult to keep all colors 'occupied'. So, if you can sacrifice a minor piece for three pawns you typically should try it, because 'three pieces less' means that the opponent will have increased problems to find occupants for all the board colors.
 
Observe that not all colors are equally dangerous!
 
The game comes with variants for five and for six colors.
The Zillions game "Deep Blue versus Kasparov" is a very special version of rainbow chess with three - blue - colors.