Book excerpt: One of the greatest problems in engineering is reliability. The performance of all machinery degrades over time and unless counteraction is taken at some point, any system will eventually fail. Once a system fails there are a number of possible solutions; the mathematical and statistical measurement and analysis of these solutions forms the mathematical theory of reliability. The aim of the authors is to concentrate on aspects of particular importance in the mathematical theory of reliability of time dependent systems rather than give a general overview. Particular emphasis is placed on fault tree analysis, Monte Carlo methods and importance measures. This book will be of particular interest to applied researchers and engineers working in areas where reliability is crucial. Contents Introduction, Markov and Semi-Markov models as a basis for the mathematical analysis of system reliability, methods for investigating homogeneous and non-homogeneous point processes (event flows), fault trees ? the current state of research, theory of redundant systems, Monte Carlo methods, reliability analysis using perturbation methods, stiff processes in reliability analysis, variance reduction methods, analytical-statistical methods for rapid simulation of repairable systems with structure redundancy, measures of reliability importance of components, index.