Integrability in a discrete world

We review our findings on integrable discrete systems with emphasis on the discrete integrability detector we have proposed under the name of singularity confinement. We have indeed shown, in a host of examples, that it is possible, by studying the structure of the singularities of discrete systems, to identify the integrable ones. A most important result of this approach is the discovery of discrete Painlevé equations of which a lengthy list exists today. These equations, being integrable systems, are characterised by particularly rich properties which are under active investigation. We present here an overview of these properties and stress the similarities and differences that exist between discrete and continuous Painlevé equations.