The theory of L2-invariants is a new and fruitful branch of mathematics, which grew up from a cooperation of algebraic topology and functional analysis and which was already successfully applied in several other branches of mathematics. The book under review represents a fundamental monograph on the theory of L2-invariants. It guides the reader from the very first definitions up to the centre of contemporary research. To a great extent, it is self-contained. Of course, the reader is assumed to have some preliminary knowledge (CW-complexes, manifolds and forms, Riemannian manifolds and sectional curvature, and some homological algebra) but the requirements are rather modest. The book is very clearly written, it contains many examples and we can find exercises at the end of each chapter. At the end of the whole book, there is an important chapter “Solutions of the Exercises”. At many places in the book, the reader will find hints for further research. In particular, at the end of each chapter, there is a section “Miscellaneous”, where it is also possible to find recommendations for further reading. Three chapters devoted to several conjectures concerning L2-invariants are especially inspiring. The book will be of great interest to specialists but it can also be strongly recommended for postgraduate students. The long list of references has 535 items.