Global properties of the space of harmonic maps from certain compact Riemann surfaces to compact Lie groups will be investigated. The method uses loop groups as symmetry groups of the harmonic map equation, to reduce the problem to consideration of an associated space of holomorphic maps. The topology of spaces of holomorphic maps from the 2-sphere to various complex manifolds will be studied by using ideas from configuration space theory, and the space of harmonic maps from a 2-torus to a compact Lie group will be studied by using methods of the theory of integrable systems.
