The purpose of this text is to offer an overview of the most popular domain decomposition methods for partial differential equations (PDE). The presentation is kept as much as possible at an elementary level with a special focus on the definitions of these methods in terms both of PDEs and of the sparse matrices arising from their discretizations. We also provide implementations written in an open source finite element software. In addition, we consider a number of methods that have not been presented in other books. We think that this book will give a new perspective and that it will complement those of Smith, Bjorstad and Gropp, Quarteroni and Valli and the comprehensive book by Toselli and Widlund as well as the review article of T. Chan. The book is addressed to computational scientists, mathematicians, physicists and, in general, to people involved in numerical simulation of partial differential equations. It can also be used as textbook for advanced undergraduate/First-Year Graduate students. The mathematical tools needed are basic linear algebra, notions of programming, variational formulation of PDEs and basic knowledge in finite element discretization. The revised and corrected version of the book is published by SIAM: http://bookstore.siam.org/ot144/ The corresponding pdf file can downloaded from Frédéric Nataf's webpage