Starting around the late 1950s, several research communities started relating the geometry of graphs to the stochastic processes on these graphs. This book, 20 years in the making, ties together the research in the field, bringing together work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written aby two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 800 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.