Discrete Painlevé equations: coalescences, limits and degeneracies

Starting from the standard form of the five discrete Painlevé equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painlevé equations. A particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n-dependence and introduces more new discrete Painlevé equations.