This book presents a very general approach to integration theory, as well as some advanced topics of the theory. It includes some new results but is also a self-contained introduction suitable for a graduate student doing self-study or an advanced course on integration theory.The book is divided into two parts. In the first part, integration theory is developed from the beginning in a general setting and for vector-valued functions which can hardly be found in other textbooks. The second part covers various topics related to integration theory, such as spaces of measurable functions, convolutions, famous paradoxes in connection with set theory, and extensions of formulas from elementary calculus to the setting of the Lebesgue integral.