This handbook on random evolutions and their applications summarizes and orders 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. It also points out some new trends. Among the subjects 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. The text also considers 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. This handbook should 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.