This volume includes some of the lectures given at two workshops, "Solitons in Physics and Mathematics" and "Solitons in Nonlinear Optics and Plasma Physics" held during the 1988-89 IMA year on Nonlinear Waves. Since their discovery by Kruskal and Zabusky in the early 1960's, solitons have had a profound impact on many fields, ranging from engineering and physics to algebraic geometry. The present contributions represent only a fraction of these areas, but give the reader a good overview of several current research directions, including optics, fluid dynamics, inverse scattering, cellular automata, Bäcklund equations, symmetries and Hamiltonian systems.