Noncommutative localization is a powerful algebraic technique for constructing new rings by inverting elements, matrices and more generally morphisms of modules. Originally conceived by algebraists (notably P.M. Cohn) it is now an important tool not only in pure algebra but also in the topology of non-simply-connected spaces, algebraic geometry and noncommutative geometry. This book is an introduction to the subject, an account of the state of the art, and also provides many references for further material. Suitable for graduate students and more advanced researchers in both algebra and topology.