1 Introduction 2 Clifford Algebras and Spin Groups 2.1 The Clifford Algebras 2.2 The groups Pin(V) and Spin(V) 2.3 Splitting of the Clifford Algebra 2.4 The complexification of the Cl(V) 2.5 The Complex Spin Representation 2.6 The Group Spine(V) 3 Spin Bundles and the Dirac Operator 3.1 Spin Bundles and Clifford Bundles 3.2 Connections and Curvature 3.3 The Dirac Operator 3.4 The Case of Complex Manifolds 4 The Seiberg-Witten Moduli Space 4.1 The Equations 4.2 Space of Configurations 4.3 Group of Changes of Gauge 4.4 The Action 4.5 The Quotient Space 4.6 The Elliptic Complex 5 Curvature Identities and Bounds 5.1 Curvature Identities 5.2 A Priori bounds 5.3 The Compactness of the Moduli Space 6 The Seiberg-Witten Invariant 6.1 The Statement 6.2 The Parametrized Moduli Space 6.3 Reducible Solutions 6.4 Compactness of the Perturbed Moduli Space 6.5 Variation of the Metric and Self-dual Two-form 6.6 Orientability of the Moduli Space 6.7 The Case when b+2 (X) > 1 6.8 An Involution in the Theory 6.9 The Case when b+(X) = 1 7 Invariants of K~hler Surfaces 7.1 The Equations over a Kahler Manifold 7.2 Holomorphic Description of the Moduli Space 7.3 Evaluation for K/ihler Surfaces 7.4 Computation for K/ihler Surfaces 7.5 Final Remarks Bibliography