This textbook will present, in a rigorous way, the basic theory of the
discrete-time and the continuous-time Markov chains, along with many
examples and solved problems. For both the topics a simple model, the
Random Walk and the Poisson Process respectively, will be used to
anticipate and illustrate the most interesting concepts rigorously
defined in the following sections. A great attention will be paid to the
applications of the theory of the Markov chains and many classical as
well as new results will be faced in the book. This textbook is intended
for a basic course on stochastic processes at an advanced undergraduate
level and the background needed will be a first course in probability
theory. A big emphasis is given to the computational approach and to
simulations.