The first edition of this book has been out of print for some years now. But as the book still seems a useful source for the main techniques, results and open problems in this area and as it has been appreciated by newcomers wanting to learn the material from scratch, there have been frequent requests for a new edition.

Since the first appearance of the book, the field has branched out in various directions: moduli spaces in positive characteristic, moduli spaces of principal bundles and of complexes, Hilbert schemes of points on surfaces, derived categories of coherent sheaves, moduli spaces of sheaves on Calabi-Yau threefolds, and many, many others. In the new appendices we make comments on some of these interesting new research directions without aiming at completeness nor going into the technical details. The main text has been left unchanged as far as possible. We used however the opportunity to correct a number of mistakes known to us and tried to improve the presentation at certain places.

We are aware of a number of excellent recent textbooks that are closely related to the topics treated here. Friedman's book [322] is a good point to start for anybody who wants to learn about vector bundles on surfaces. It combines an introduction to the theory of algebraic surfaces with a study of vector bundles. Moduli spaces of vector bundles on curves are constructed and discussed in Le Potier's book [367].