Markov chains are an important idea, related to random walks, which crops up widely in applied stochastic analysis. They are used for example in performance modeling and evaluation of computer networks, queuing networks, and telecommunication systems. The main point of the present book is to provide methods, based on the construction of Lyapunov functions, of determining when a Markov chain is ergodic, null recurrent, or transient. These methods, which are on the whole original and new, can also be extended to the study of questions of stability. Of particular concern are reflected random walks and reflected Brownian motion. Here, the authors provide a self-contained introduction to the theory and details of how the required Lyapunov functions are constructed in various situations.