this book applies the recent techniques of gauge theory to study the smooth classification of compact complex surfaces. the study is divided into four main areas: classical complex surface theory, gauge theory and donaldson invariants, deformations of holomorphic vector bundles, and explicit calculations for elliptic sur§ faces. the book represents a marriage of the techniques of algebraic geometry and 4-manifold topology and gives a detailed exposition of some of the main themes in this very active area of current research.