The topic of this book is the theory of semistable coherent sheaves on a smooth algebraic surface and of moduli spaces of such sheaves. The content ranges from the definition of a semistable sheaf and its basic properties over the construction of moduli spaces to the birational geometry of these moduli spaces. The book is intended for readers with some background in Algebraic Geometry, as for example provided by Hartshorne's textbook [98].

There are at least three good reasons to study moduli spaces of sheaves on surfaces. Firstly, they provide examples of higher dimensional algebraic varieties with a rich and interesting geometry. In fact, in some regions in the classification of higher dimensional varieties the only known examples are moduli spaces of sheaves on a surface. The study of moduli spaces therefore sheds light on some aspects of higher dimensional algebraic geometry. Secondly, moduli spaces are varieties naturally attached to any surface. The understanding of their properties gives answers to problems concerning the geometry of the surface, e.g. Chow group, linear systems, etc. From the mid-eighties till the mid-nineties most of the work on moduli spaces of sheaves on a surface was motivated by Donaldson's ground breaking results on the relation between certain intersection numbers on the moduli spaces and the differentiable structure of the four-manifold underlying the surface. Although the interest in this relation has subsided since the introduction of the extremely powerful Seiberg-Witten invariants in 1994, Donaldson's results linger as a third major motivation in the background; they throw a bridge from algebraic geometry to gauge theory and differential geometry.