Discrete Painleve equations: an integrability paradigm

In this paper we present a review of results on discrete Painleve equations. We begin with an introduction which serves as a refresher on the continuous Painleve equations. Next, in the first, main part of the paper, we introduce the discrete Painleve equations, the various methods for their derivation, and their properties as well as their classification scheme. Along the way we present a brief summary of the two major discrete integrability detectors and of Quispel-Roberts-Thompson mapping, which plays a primordial role in the derivation of discrete Painleve equations. The second part of the paper is more technical and focuses on the presentation of new results on what are called asymmetric discrete Painleve equations.