Derived Equivalences for Group Rings A self contained introduction is given to J. Rickards Morita theory for derived module categories and its recent applications in representation theory of finite groups. In particular Broués conjecture is discussed giving a structural explanation for relations between the p modular character table of a finite group and that of its p local structure . The book is addressed to researchers or graduate students and can serve as material for a seminar. It surveys the current state of the field and it also provides a users guide to derived equivalences and tilting complexes. Results and proofs are presented in the generality needed for group theoretic applications