Topological solitons have been investigated by theoretical physicists and mathematicians for more than a quarter of a century, and it is now a good time to survey the progress that has been made. Many types of soliton have been understood in detail, both analytically and geometrically, and also numerically, and various links between them have been discovered.

This book introduces the main examples of topological solitons in classical field theories, discusses the forces between solitons, and surveys in detail both static and dynamic multi-soliton solutions. Kinks in one dimension, lumps and vortices in two dimensions, monopoles and Skyrmions in three dimensions, and instantons in four dimensions, are all discussed. In some field theories, there are no static forces between solitons, and there is a large class of static multi-soliton solutions satisfying an equation of the Bogomolny type. Deep mathematical methods can be used to investigate these. The manifold of solutions is known as moduli space, and its dimension increases with the soliton number. We survey the results in this area. We also discuss the idea of geodesic dynamics on moduli space, which is an adiabatic theory of multi-soliton motion at modest speeds when the static forces vanish, or almost vanish.

Some variants of the solitons mentioned above are considered, but we do not consider the coupling of fermions to solitons, nor solitons in supersymmetric theories, where there are sometimes remarkable dualities between the solitons and elementary particles, nor solitons coupled to gravity, although all these topics are interesting.