At first there was the Markov property. The theory of shastic processes, which can be considered as an extension of probability theory, allows the modeling of the evolution of systems through the time. It cannot be properly understood just as pure mathematics, separated from the body of experience and examples that have brought it to life. The theory of shastic processes entered a period of intensive development, which is not finished yet, when the idea of the Markov property was brought in. Not even a serious study of the renewal processes is possible without using the strong tool of Markov processes. The modern theory of Markov processes has its origins in the studies by A. A: Markov (1856-1922) of sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian motion. Later, many generalizations (in fact all kinds of weakenings of the Markov property) of Markov type shastic processes were proposed. Some of them have led to new classes of shastic processes and useful applications. Let us mention some of them: systems with complete connections [90, 91, 45, 86]; K-dependent Markov processes [44]; semi-Markov processes, and so forth. The semi-Markov processes generalize the renewal processes as well as the Markov jump processes and have numerous applications, especially in reliability.