This textbook explores deeply the theory of measure and integration through examples, counterexample, exercises and problem solving. As a first course, the book is designed to provide a self-contained document for graduate students and researchers involved in analysis and probability. The general topics covered by the book are indispensable in understanding most of concepts in functional analysis, harmonic analysis, probability theory, stochastic analysis.The book does not present the theory of measure and integration only through theorems and proofs. Instead, some important concepts and selected themes are proposed within exercises and problems. This approach is chosen purposely.A collection of 140 exercises and 20 problems are set out at the end of each chapter with complete solutions. The second part of the textbook offers fully detailed solutions of the proposed problems and exercises.The challenge was to make the text accessible for any student with a modest background in real analysis without missing the basics and essential results of the theory of measure and integration.Further to the theory, the textbook offers a large numbers of exercises and problems with detailed solutions. Most solved problems tackle some specific topic not covered by the theoretical part.The book is practical both as a teaching document for Bachelor, Master students, and first-year PhD students. It is also suitable for advanced undergraduate students and lecturers.