Open Quantum Systems : a mathematical perspective This book presents four survey articles on various aspects of open quantum systems specifically addressing quantum Markovian processes Feller semigroups and nonequilibrium dynamics. The contributions are based on lectures given by distinguished experts at a summer school in Göttingen Germany. Starting from basic notions the authors of these lecture notes accompany the reader on a journey up to the latest research highlighting new challenges and addressing unsolved problems at the interface between mathematics and physics. Though the book is primarily addressed to graduate students it will also be of interest to researchers Introduction to Classical and Quantum Markov Semigroups Introduction to Non Markovian Evolution of n Level Quantum Systems Aspects of Micro local Analysis and Geometry in the Study of Lévy type Generators Lectures on Entropy. I: Information Theoretic Notions