Large deviations theory is a branch of probability theory that studies the asymptotic behavior of rare events, particularly the exponential decay of their probabilities. This book provides a rigorous exposition of large deviations theory, covering both the theoretical foundations and practical applications. The book is structured into several chapters, each focusing on different aspects of large deviations, including finite-dimensional spaces, Markov chains, random walks, and projective limits. It also includes a detailed discussion of the Gibbs conditioning principle, the Gärtner–Ellis theorem, and concentration inequalities. The book is intended for a broad audience, ranging from senior undergraduate students to advanced graduate students in mathematics, statistics, and engineering. The text includes exercises, historical notes, and references to help readers understand the subject. The book has been revised and updated in its second edition, incorporating new material and corrections. The authors have also provided a web site with additional information and corrections related to the second edition. The book is an essential resource for anyone interested in large deviations theory and its applications in various fields.Large deviations theory is a branch of probability theory that studies the asymptotic behavior of rare events, particularly the exponential decay of their probabilities. This book provides a rigorous exposition of large deviations theory, covering both the theoretical foundations and practical applications. The book is structured into several chapters, each focusing on different aspects of large deviations, including finite-dimensional spaces, Markov chains, random walks, and projective limits. It also includes a detailed discussion of the Gibbs conditioning principle, the Gärtner–Ellis theorem, and concentration inequalities. The book is intended for a broad audience, ranging from senior undergraduate students to advanced graduate students in mathematics, statistics, and engineering. The text includes exercises, historical notes, and references to help readers understand the subject. The book has been revised and updated in its second edition, incorporating new material and corrections. The authors have also provided a web site with additional information and corrections related to the second edition. The book is an essential resource for anyone interested in large deviations theory and its applications in various fields.