Absorbing Markov chains provide a versatile tool to describe the condition of a system subject to failure. Absorption meaning failure of the system leads to phase-type distributed times to failure. The aim of this book is to give an appropriate maintenance model for systems with phase-type distributed times to failure. The main part of the book covers the description of the user''s opportunities of interaction. Having a failure, the system may be repaired, and while the system is working, we may perform a preventive maintenance action. The maintenance actions are represented by stochastic matrices which, as well as the intensity matrix of the Markov chain, are the essential mathematical objects investigated. We also try to give a satisfying explanation to the question how to choose an optimal maintenance policy. The optimality criterion considered is to maximize the expected gains rewarded by the system. We discuss four approaches for solving the maximization problem and we finish with a discussion of possible extensions of the model and subjects that may interest further research about this model.