The area of Markov chain theory and application has matured over the past 20 years. This publication deals with the action of Markov chains on general state spaces. It discusses the theories and the use to be gained, concentrating on the areas of engineering, operations research and control theory. Throughout, the theme of stochastic stability and the search for practical methods of verifying such stability, provide a new and powerful technique which not only affects applications, but also the development of the theory itself. The impact of the theory on specific models is discussed in detail.