This book is based upon lectures delivered during the Summer School on Symmetries and Integrability of Difference Equations at the Université de Montréal, Canada, June 8, 2008–June 21, 2008. The lectures are devoted to methods that have been developed over the last 15–20 years for discrete equations. They are based on either the inverse spectral approach or on the application of geometric and group theoretical techniques. The topics covered in this volume can be summarized in the following categories:

• Integrability of difference equations

• Discrete differential geometry

• Special functions and their relation to continuous and discrete Painlevé functions

• Discretization of complex analysis

• General aspects of Lie group theory relevant for the study of difference equations. Specifically, two such subjects are treated: 1. Cartan's method of moving frames 2. Lattices in Euclidean space, symmetrical under the action of semisimple Lie groups

• Lie point symmetries and generalized symmetries of discrete equations

Twelve distinct lecture series were presented at the Summer School of which eleven are included in this volume. Close to 50 registered graduate students and researchers from twelve different countries participated.

The Summer School, Séminaire de mathématiques supérieures, is a yearly event at the Département de Mathématiques, Université de Montréal. The organizing committee for the year 2008 consisted of Pavel Winternitz (Université de Montréal, Canada), Vladimir Dorodnitsyn (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences), Decio Levi (Università degli Studi Roma Tre, Italy) and Peter Olver (University of Minnesota, USA).