Semiclassical methods based on classical solutions play an important role in quantum field theory, high energy physics, and cosmology. Real-time soliton solutions give rise both to new particles, such as magnetic monopoles, and to extended structures, such as domain walls and cosmic strings. These could have been produced as topological defects in the very early universe. Confronting the consequences of such objects with observation and experiment places important constraints on grand unification and other potential theories of high energy physics beyond the standard model. Imaginary-time Euclidean instanton solutions are responsible for important nonperturbative effects. In the context of quantum chromodynamics they resolve one puzzle—the U(1) problem—while raising another—the strong CP problem—whose resolution may entail the existence of a new species of particle, the axion. The Euclidean bounce solutions govern transitions between metastable vacuum states. They determine the rates of bubble nucleation in cosmological first-order transitions and give crucial information about the evolution of these bubbles after nucleation. These bounces become of particular interest if there is a string theory landscape with a myriad of metastable vacua.

This book is intended as a survey and overview of this field. As the title indicates, there is a dual focus. On the one hand, solitons and instantons arise as solutions to classical field equations. The study of their many varieties and their mathematical properties is a fascinating subfield of mathematical physics that is of interest in its own right. Much of the book is devoted to this aspect, explaining how the solutions are discovered, their essential properties, and the topological underpinnings of many of the solutions. However, the physical significance of these classical objects can only be fully understood when they are seen in the context of the corresponding quantum field theories. To that end, there is also a discussion of quantum effects, including those arising from the interplay of fermion fields with topologically nontrivial classical solutions, and of some of the phenomenological consequences of instantons and solitons.