This is the first handbook on random evolutions and their applications. Its main purpose is to summarize and order the ideas, methods, results and literature on the theory of random evolutions since 1969 and their applications to the evolutionary stochastic systems in random media, and also to point out some new trends. Among the subjects that are treated are the problems for different models of random evolutions, multiplicative operator functionals, evolutionary stochastic systems in random media, averaging, merging, diffusion approximation, normal deviations, rates of convergence for random evolutions and their applications. New developments, such as the analogue of Dynkin's formula, boundary value problems, stability and control of random evolutions, stochastic evolutionary equations, driven space-time white noise and random evolutions in financial mathematics are also considered. Audience: This handbook will be of use to theoretical and practical researchers whose interests include probability theory, functional analysis, operator theory, optimal control or statistics, and who wish to know what kind of information is available in the field of random evolutions and their applications.