This book presents a general approach to integration theory, as well as some advanced topics. It includes some new results, but is also a self-contained introduction suitable for a graduate student doing self-study or for an advanced course on integration theory.The book is divided into two parts. In the first part, integration theory is developed from the start in a general setting and immediately for vector-valued functions. This material 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, and extensions of formulae from elementary calculus to the setting of the Lebesgue integral.