The geometric approach to quantization was introduced by Kostant and Souriau more that twenty years ago. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. The ideas have also proved useful in pure mathematics, noteably in representation theory, with the most recent applications occurring in conformal field theory and in the Jones-Witten theory of knots. For the second edition of this successful monograph it has been completely revised and the presentation has been simplified. A large number of examples have been added and the material on field theory has been expanded.