This volume is the second and substantially enlarged edition of the book that appeared for the first time in 1984. K. Hulek has joined the authors of the first edition as a fourth co-author. During the last two decades, there have been several interesting developments in the theory of complex surfaces and they are reflected in this new edition. There are new sections in chapter 4 (including a discussion of Kähler structures on surfaces, a treatment of pluricanonical maps of surfaces based on the Reider theorem, Bogomolov's stability for rank 2 vector bundles and an introduction to nef-divisors) and chapter 8 (mirror symmetry for projective K3-surfaces, special curves on K3-surfaces and applications to hyperbolic geometry). Chapter 9 is an important addition, which is devoted to a study of topological and differentiable structures on surfaces. It contains an introduction to the Donaldson and the Seiberg-Witten invariants. The bibliography has been substantially extended, covering new developments. Already a classic in the field, this book is recommended both to mathematicians interested in this mathematical topic and physicists working in the modern theoretical physics.