This book is devoted to a topic that has undergone rapid and fruitful development over the last few years: symmetries and integrability of difference equations and $q$-difference equations and the theory of special functions that occur as solutions of such equations. Techniques that have been traditionally applied to solve linear and nonlinear differential equations are now being successfully adapted and applied to discrete equations.This volume is based on contributions made by leading experts in the field during the workshop on Symmetries and Integrability of Difference Equations held in Esterel, Quebec, in May 1994. Giving an up-to-date review of the current status of the field, the book treats these specific topics: Lie group and quantum group symmetries of difference and $q$-difference equations, integrable and nonintegrable discretizations of continuous integrable systems, integrability of difference equations, discrete Painleve property and singularity confinement, integrable mappings, applications in statistical mechanics and field theories, Yang-Baxter equations, $q$-special functions and discrete polynomials, and $q$-difference integrable systems.