We present a study of semi-Markov random evolutions in discrete-time and study asymptotic properties, namely, averaging and diffusion approximation by martingale weak convergence method. We start from a short presentation of semi-Markov chains, the Markov renewal theory and then the definition of semi-Markov random evolution and limit theorems followed by applications in concrete families of stochastic systems. Applications concern averaging and diffusion approximations for discrete-time dynamical systems, additive functionals, geometric Markov renewal processes. We also present some estimation problem of stationary probability of semi-Markov chain.
