I. Introduction.- Solitons in Mathematics: Brief History.- Solitons in Physics.- II. Mathematical Aspects.- Numerical Studies of Solitons.- Poles of the Toda Lattice.- Perturbation Theory of the Double Sine-Gordon Equation.- Soliton Perturbations and Nonlinear Focussing.- Novel Class of Nonlinear Evolution Equations Solvable by the Spectral Transform Technique, Including the So-Called Cylindrical KdV Equation.- The Complex Modified Korteweg-de Vries Equation, a Non-Integrable Evolution Equation.- III. Statistical Mechanics and Solid-State Physics.- Soliton-Bound States in the Magnetic Gap.- Statistical Mechanics of Nonlinear Dispersive Systems.- Some Applications of Instantons in Statistical Mechanics.- The Theory of Structural Phase Transitions: Cluster Walls and Phonons.- Nonlinear Lattice Dynamics: Molecular Dynamics Studies.- Computer Simulation of Structural Phase Transitions.- Soliton-Like Features in a Two-Dimensional XY Model with Ouartic Anisotropy.- Behavior of a ?4-Kink in the Presence of an Inhomogeneous Perturbation.- Solitary Wave Solutions in a Diatomic Lattice.- Lattice Models of High Velocity Dislocation Motion.- Grain Boundaries as Solitary Waves.- The Relation of Solitons to Polaritons in Coupled Systems.- Solitons in CsNiF3: Their Experimental Evidence and Their Thermodynamics.- Structure and Stability of Domain Walls - Phase Transition.- Periodic Lattice Distortions and Charge Density Waves in One- and Two-Dimensional Systems.- Solitons in Incommensurate Systems.- Fluctuations and Freezing in a One-Dimensional Liquid: Hg3??AsF6.- Charge Density Wave Systems: The ?-Particle Model.- The Soliton Lattice: Application to the ? Phase.- The New Concept of Transitions by Breaking of Analyticity in a Crystallographic Model.- Textures in Superfluid 3He.- Creation of Spin Waves in 3HeB.- The Interaction of Spin Waves in Liquid He3 in Several Dimensions.- Josephson Transmission Line Oscillators.- Dissipative Structures in Quasi-One-Dimensional Superconductors.- Solitary Phenomena in Finite Dissipative Discrete Systems.- Stability of Nonuniform States in Systems Exhibiting Continuous Bifurcation.- The Sine-Gordon Chain: Mass Diffusion.- Solitary-Wave Propagation as a Model for Poling in PVF2.- Theory of One-Dimensional Ionic and Solitary Wave Conduction in Potassium Hollandite.- IV. Summary.- Summary: Where Do Solitons Go From Here?.- Index of Contributors.