This chapter introduces Markov chain Monte Carlo (MCMC) methods—a powerful class of algorithms for sampling from complex probability distributions by constructing Markov chains with the desired stationary distribution. After reviewing the fundamental concepts of Markov chains, including stationarity, irreducibility, and detailed balance, the chapter presents the Metropolis, Metropolis–Hastings, and Gibbs sampling algorithms with practical guidance for their construction and implementation. Applications to classical problems such as graph coloring, hard-core models, the Ising model, and approximate counting in combinatorial problems are discussed, along with an introduction to perfect simulation techniques and an example from computational biology: the motif-finding problem. The chapter also addresses stable simulation schemes and practical criteria for assessing convergence in simulations. Throughout, step-by-step algorithms, illustrative figures, and publicly available Python code are provided to support understanding and practical use. By the end of the chapter, readers will be equipped with both the theoretical insight and practical tools to apply MCMC methods in a range of scientific and statistical applications.