Discrete Painleve equations: A review

We present a review of what is current knowledge about discrete Painleve equations. We start with a historical introduction which explains how the discrete Painleve equations made their appearance. Since the most widely used derivation method was the one based on integrability detectors, we start by presenting a brief review of the existing discrete integrability detectors, and proceed to show how they were indeed used in deriving the discrete Painleve equations. Given the profusion of different forms of these mappings, we examine the various approaches to their classification, from the first, crude ones, to the complete classification based on affine Weyl groups. The properties of the discrete Painleve equations, which make them so special, are also reviewed with ample details. Finally, we present a series of results which are peripherally related to the discrete Painleve equations but help cast more light on these (very) special systems.