The Markov processes are an important class of the stochastic processes. The Markov property means that evolution of the Markov process in the future depends only on the present state and does not depend on past history. The Markov process does not remember the past if the present state is given. Hence, the Markov process is called the process with memoryless property. This chapter covers some basic concepts, properties, and theorems on homogeneous Markov chains and continuous-time homogeneous Markov processes with a discrete set of states. The theory of those kinds of processes allows us to create models of real random processes, particularly in issues of reliability and maintenance.