The last ten years have seen rapid advances in the understanding of differentiable four-manifolds, not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics, and have proven to be a mainspring of current mathematical research. This book provides a lucid and accessible account of the modern study of the geometry of four-manifolds. Consequently, it will be required reading for all those mathematicians and theoretical physicists whose research touches on this topic. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology--notably the definition of new invariants of four-manifolds--and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proven in detail. On the geometric side, the book contains a new proof of the classifica