Nonlinear equations come to us in tremendous variety, each with its own questions and its own difficulties. At one extreme are the completely integrable equations, with constants of the motion and a rich algebraic structure. At the other extreme is chaos, with turbulent solutions and statistical averages. Between these two possibilities, algebraic and ergodic, lies the full range of nonlinear phenomena. There are smooth solutions which develop shocks, or bifurcate, or maintain slow and nearly periodic variations that imitate the linear theory. Each of these questions requires a separate treatment, and the subject would be simpler if we know for every equation which behavior to expect. Nevertheless these equations, the nonlinear partial differential equations which arise in applications, share one crucial property. They are all vulnerable when the right pattern in found. It is a slow process, to uncover and reveal their structure, but it is moving forward. The papers in this volume reflect a part of that progress. They were presented at the U.S.-Japan Seminar in Tokyo in July 1982.