This is a research monograph on soliton solutions of elliptic partial differential equations arising in field theory, solutions such as vortices, instantons, monopoles, dyons, and cosmic strings. The book presents in-depth descriptions of important problems of current interest from several major branches of field theory, including electromagnetism, superconductivity, Yang-Mills gauge theory, and cosmology. It forces a link between mathematical analysis and physics and seeks to stimulate further research in the area. Mathematically, the book involves Riemannian geometry, Lie groups and Lie algebras, and algebraic topology (characteristic classes and homotopy), and emphasizes modern nonlinear functional analysis. While field theory has long been of interest to algebraists, geometers, and topologists, it has gradually begun to attract the attention of more analysts. Written for researchers and graduate students, this book will appeal to mathematicians and theoretical physicists.