The evolution of systems is a growing field of interest stimulated by many possible applications. This work is devoted to semi-Markov random evolutions (SMRE). This class of evolutions is rich enough to describe the evolutionary systems changing their characteristics under the influence of random factors. At the same time there exist efficient mathematical tools for investigating the SMRE. The topics addressed in this book include classification, fundamental properties of the SMRE, averaging theorems, diffusion approximation and normal deviations theorems for SMRE in ergodic case and in the scheme of asymptotic phase lumping. Both analytic and stochastic methods for investigation of the limiting behaviour of SMRE are developed. This text includes many applications of rapidly changing semi-Markov random media, including storage and traffic processes, branching and switching processes, stochastic differential equations, motions on Lie groups, and harmonic oscillations. This volume should be of interest to statisticians whose work involves operator theory, functional analysis, systems theory and cybernetics.