The geometric approach to quantization was introduced by Kostant and Souriau more than twenty years ago. It has given valuable and lasting insights into the relationship between classical and quantum systems. It continues to be a popular research topic. The ideas have proved useful in pure mathematics, notably in representation theory, as well as in theoretical physics. The most recent applications have been in conformal field theory and in the Jones-Witten theory of knots. The original edition of this book was published in 1980. For this edition it has been completely revised and extensively rewritten. The presentation has been simplified and a large number of examples have been added. The material on field theory has been expanded.