The Hunting for the Discrete Painleve Equations

We review our findings on discrete Painleve equations with emphasis on the two direct methods we have proposed for their derivation: deautonomisation through singularity confinement and geometrical approach based on affine Weyl groups. The question of integrability of discrete Painleve equations is also addressed in terms of the existence of a Lax pair or of a description in the frame of the Grand Scheme. A list of discrete Painleve equations, as complete as possible but still not exhaustive, is also presented.