This tutorial by Christophe Andrieu and Johannes Thoms provides a comprehensive review of adaptive Markov chain Monte Carlo (MCMC) algorithms, focusing on optimizing their performance. The authors use simple toy examples to explore the theoretical foundations of adaptive MCMC, highlighting why these algorithms can fail under certain conditions. They provide guidelines for designing correct algorithms and introduce the framework of stochastic approximation, which allows for systematic optimization of commonly used criteria. The tutorial also presents novel adaptive algorithms that are robust and reliable in practice, demonstrating their effectiveness through artificial and high-dimensional scenarios, as well as a classic mine disaster dataset inference problem. The introduction explains the basic principles of MCMC, emphasizing the importance of the Metropolis-Hastings (MH) algorithm and the role of proposal distributions in generating Markov chains. It discusses the challenges posed by the choice of proposal distributions and the need for efficient and reliable MCMC algorithms. The tutorial then delves into the theoretical underpinnings of controlled Markov chains, a technique used to optimize MCMC transition probabilities, and introduces the concept of stochastic approximation for sequential updating of parameters. The authors aim to provide a clear and practical guide to optimizing MCMC algorithms, offering new insights and methods for improving their performance.This tutorial by Christophe Andrieu and Johannes Thoms provides a comprehensive review of adaptive Markov chain Monte Carlo (MCMC) algorithms, focusing on optimizing their performance. The authors use simple toy examples to explore the theoretical foundations of adaptive MCMC, highlighting why these algorithms can fail under certain conditions. They provide guidelines for designing correct algorithms and introduce the framework of stochastic approximation, which allows for systematic optimization of commonly used criteria. The tutorial also presents novel adaptive algorithms that are robust and reliable in practice, demonstrating their effectiveness through artificial and high-dimensional scenarios, as well as a classic mine disaster dataset inference problem. The introduction explains the basic principles of MCMC, emphasizing the importance of the Metropolis-Hastings (MH) algorithm and the role of proposal distributions in generating Markov chains. It discusses the challenges posed by the choice of proposal distributions and the need for efficient and reliable MCMC algorithms. The tutorial then delves into the theoretical underpinnings of controlled Markov chains, a technique used to optimize MCMC transition probabilities, and introduces the concept of stochastic approximation for sequential updating of parameters. The authors aim to provide a clear and practical guide to optimizing MCMC algorithms, offering new insights and methods for improving their performance.