This is a book on the Reidemeister torsion and its (mostly Turaev’s) generalizations. When comparing it with Turaev’s book (Introduction to Combinatorial Torsions, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2001), which appeared recently, we can immediately see that it has a different character. While Turaev’s book can also serve as a kind of introduction to the subject (as well as an introduction to the contemporary research in the field), the book under review is devoted to a wide range of applications of the torsion, and its reading requires certain prerequisites. In the first chapter, which makes the reader familiar with the necessary algebraic notions, the reader is supposed to have some topological (and also algebraic) background. The author modestly states in the introduction that this is a computationally oriented little book. But let us note that the “computations” we find here are very clever computations, and the wide variety of applications presented here do not support the description of a little book. The book will be indispensable for specialists in the field, and I think that it is very good that it exists together with Turaev’s book. It is very well written, with many examples and also many exercises. The wide range of applications will be interesting not only for topologists but also for differential geometers.