This book is designed to introduce graduate students and researchers to the primary methods useful for approximating integrals. The emphasis is on those methods that have been found to be of practical use, and although the focus is on approximating higher-dimensional integrals the lower-dimensional case is also covered. This book covers all the most useful approximation techniques so far discovered; the first time that all such techniques have been included in a single book and at a level accessible for students. In particular, it includes a complete development of the material needed to construct the highly popular Markov Chain Monte Carlo (MCMC) methods.