This volume collects selected contributions from the workshop on random walks, which was held at the Schrödinger Institute in Vienna in 2001. Topics of this special semester included probability and random walks on groups and graphs, products of random matrices, potential theory, Brownian motion on manifolds, combinatorial and spectral properties of random walks on graphs and fractals. The volume concentrates on geometric applications like hyperbolic geometry, Lie groups, and geometric group theory. The book is divided into two parts, longer surveys and articles (pp. 1-295) followed by shorter research communications (pp. 296-532). Some additional keywords of the main contributions to this book are Lyapunov spectrum, stochastic Loewner equation, expander graphs, quantum chaos, cellular automata and symbolic dynamical systems. The volume is dedicated to the memory of Martine Babillot.