Markov chains are among the basic and most important examples of random processes. This book is about timehomogeneous Markov chains that evolve with discrete time steps on a countable state space. A specific feature is the systematic use, on a relatively elementary level, of generating functions associated with transition probabilities for analyzing Markov chains. Basic definitions and facts include the construction of the trajectory space and are followed by ample material concerning recurrence and transience, the convergence and ergodic theorems for positive recurrent chains. There is a sidetrip to the PerronFrobenius theorem. Special attention is given to reversible Markov chains and to basic mathematical models of population evolution such as birthanddeath chains, GaltonWatson process and branching Markov chains. A good part of the second half is devoted to the introduction of the basic language and elements of the potential theory of transient Markov chains. Here the construction and properties of the Martin boundary for describing positive harmonic functions are crucial. In the long final chapter on nearest neighbor random walks on (typically infinite) trees the reader can harvest from the seed of methods laid out so far, in order to obtain a rather detailed understanding of a specific, broad class of Markov chains. The level varies from basic to more advanced, addressing an audience from master's degree students to researchers in mathematics, and persons who want to teach the subject on a medium or advanced level. Measure theory is not avoided; careful and complete proofs are provided. A specific characteristic of the book is the rich source of classroomtested exercises with solutions. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in Markov chains. Table of Contents  Preliminaries and basic facts
 Irreducible classes
 Recurrence and transience, convergence, and the ergodic theorem
 Reversible Markov chains
 Models of population evolution
 Elements of the potential theory of transient Markov chains
 The Martin boundary of transient Markov chains
 Minimal harmonic functions on Euclidean lattices
 Nearest neighbour random walks on trees
 Solutions of all exercises
 Bibliography
 List of symbols and notation
 Index
