This book is based on lectures given by the authors at an instructional conference on integrable systems held at the Mathematical Institute in Oxford in September 1997. Most of the participants were graduate students from the United Kingdom and other European countries. The lectures emphasized geometric aspects of the theory of integrable systems, particularly connections with algebraic geometry, twistor theory, loop groups, and the Grassmannian picture. We are grateful for support for the conference from the London Mathematical Society, the Engineering and Physical Sciences Research Council (contract No. 00985SCI96), the University of Oxford Mathematical Prizes Fund, the Mathematical Institute, Wadham College, and Oxford University Press.