The method of inverse scattering transformation and the theory of solitons are among the most recent and fascinating achieve- ments in the domain of applied mathematics. The development of the theory, which took place mainly in the last decade, has been explosive and far-reaching. It is not our ambition to cover the whole field in this book. Our aim, in the first place, is to provide an introduction for the uninitiated reader, for the mathematician or physicist who has never heard about solitons, or witnessed only from some distance the excitement that they have created. Secondly however, we have made it our goal to present the theory in a mathematically satisfactory and rigorous way. As a result the book contains various new points of view, and material that cannot be found elsewhere. This, we hope, will also be of interest to scientists who are experienced in the soliton theory.