This is a survey article which explains how the theory of integrable systems, in particular constructions related to harmonic maps into symmetric spaces, can be used to study many problems in geometry. It begins with the classical Enneper-Weierstrass representation of minimal surfaces, then generalizes, using loop groups, to the infinite-dimensional analogue for the non-minimal constant mean curvature surfaces of J. F. Dorfmeister, F. J. Pedit and H. Wu, and ends with work of the authors on Hamiltonian stationary Lagrangian surfaces in 2-dimensional complex symmetric spaces.