This book starts with chapters on classical notions of integration and measure theory (the Riemann integral, the Lebesgue measure, the Carathéodory process, product measures, abstract integration, Lp-spaces, Radon and Hausdorff measures, the Vitali, Besicovitch and Rademacher theorems and maximal functions). The next chapters explore connections between measure theory and probability theory (ergodic theory, laws of large numbers, the central limit theorem, the Wiener measure, Brownian motion and martingales). The book has several appendices on topological notions, diffeomorphisms, the Whitney extension theorem, the Marcinkiewicz interpolation theorem, Sard's theorem, integration of differential forms and the Gauss-Green formula. Each chapter ends with many exercises of varying difficulty, which give further applications and extensions of the theory. The book ends with a bibliography, a list of standard symbols and a subject index. The book is very understandable, requiring only a basic knowledge of analysis. It can be warmly recommended to a broad spectrum of readers, to graduate students as well as young researchers.