The purpose of this article is to present the semi-Markov processes focusing on applications, especially in reliability and dependability. The under consideration semi-Markov processes are both of continuous and discrete time with countable or finite state space. The basic definitions of Markov renewal and semi-Markov processes are presented, as well as the Markov renewal theorem and the necessary theory of statistical estimation. In the sequel, we describe a general reliability model and give

more »

... he corresponding estimations. Finally, we briefly present the semi-Markov chain and hidden semi-Markov models. Some bibliographical directions for further relevant topics, which are not included in this article, are given.