Reinvigorated by advances and insights, in particular from the active fields of quantum information and computing, the quantum theory of irreversible processes has recently attracted growing attention. This volume introduces the very basic concepts of semigroup dynamics of open quantum systems and reviews a variety of modern applications. The present book, originally published as Vol. 286 (1987) in Lecture in Physics, has been newly typeset, revised and corrected and also expanded to include a review on recent developments.