Explains the ideas involved in applying the theory of integrable systems to finding harmonic maps and related geometric objects. Harmonic maps, which are maps between Riemannian or pseudo- Riemannian manifolds that extremize a natural energy integral, are applied in the theory of minimal an constant mean curvature surfaces, and as the non-linear sigma and chiral models of particle physics. An introduction for graduate students and mathematicians, and suitable as a supplement or core text for a graduate course.