Main subject categories: • Markov processes • Discrete-time Markov processes • Continuous-time Markov processes • Markov chains • AnalysisMathematics Subject Classification (2010): • 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) • 60J27 Continuous-time Markov processes on discrete state spaces • 60J28 Applications of continuous-time Markov processes on discrete state spaces • 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.