The core of this book is a one-year course in probability theory and the theory of random processes, taught at Princeton University to undergraduate and graduate students. The book provides a comprehensive, self-contained exposition of classical probability theory and the theory of random processes, and dwells on a number of modern topics not addressed in most textbooks. There is detailed discussion of Lebesgue integration, Markov chains, random walks, laws of large numbers, limit theorems, and their relation to Renormalization Group theory. It also includes the theory of stationary random processes, martingales, generalized random processes, and Brownian motion. One section is devoted to the theory of Gibbs random fields. The material is essential to many undergraduate and graduate courses. The book can also serve as a reference for scientists using modern probability theory in their research.