Topological Aspects of Four-Manifolds -- The Theory of Donaldson Invariants -- The Theory of Seiberg-Witten Invariants -- Supersymmetry in Four Dimensions -- Topological Quantum Field Theories in Four Dimensions -- The Mathai-Quillen Formalism -- The Seiberg-Witten Solution of N = 2 SUSY Yang-Mills Theory -- The u-plane Integral -- Some Applications of the u-plane Integral -- Further Developments in Donaldson—Witten Theory.The present book is the first of its kind in dealing with topological quantum field theories and their applications to topological aspects of four manifolds. It is not only unique for this reason but also because it contains sufficient introductory material that it can be read by mathematicians and theoretical physicists. On the one hand, it contains a chapter dealing with topological aspects of four manifolds, on the other hand it provides a full introduction to supersymmetry. The book constitutes an essential tool for researchers interested in the basics of topological quantum field theory, since these theories are introduced in detail from a general point of view. In addition, the book describes Donaldson theory and Seiberg-Witten theory, and provides all the details that have led to the connection between these theories using topological quantum field theory. It provides a full account of Witten’s magic formula relating Donaldson and Seiberg-Witten invariants. Furthermore, the book presents some of the recent developments that have led to important applications in the context of the topology of four manifolds.