This book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. One of the many consequences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to 4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These results have had far-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.