This monograph provides a significantly revised and expanded version of the original edition published in 1987 on the classical topics in measure theory and integration. The book starts with chapters on classical notions of measure theory (measurability, measure, measurable functions including sections on Lebesgue--Stieltjes measures, Carathéodory process, metric outer measures, or convergences of measures) and integration (abstract Lebesgue integration, convergence theorems, the Lp-spaces and the Vitali-Hahn-Saks theorem). Next chapters are devoted to differentiation and duality (the Hahn decomposition, the Radon-Nikodým theorem, dual spaces) and product measures and integrals. The material in the new chapter is concerned with the Henstock-Kurzweil nonabsolute integration, both on the real line and in Banach spaces.It follows an interesting chapter on analytic sets and Choquet's theory of capacities with an application to the Daniell integral. Last chapters treat materials on an elementary proof of the lifting theorem and on topological measures (the Riesz-Markov theorem, Haar measures). The final chapter is devoted to relations between abstract and topological measures. Each chapter is companioned by many exercises of varying difficulty which give further applications and extensions of the theory. The book is equipped with a bibliography (more than 170 references), list of standard symbols and notation and with the author and subject indices. The book is well understandable and requires only a basic knowledge of advanced calculus; moreover, some results from topology and set theory are collected in a short appendix, and also a section on basic cornerstones of Banach spaces is included. It can be warmly recommended to a broad spectrum of readers--to graduate students as well as young researches who wish to become acquainted with the basic elements and deeper properties of abstract analysis and its applications.