  The configuration space of a non-linear sigma model is the space of maps from
one manifold to another. This paper reviews the authors' work on non-linear
sigma models with target a homogeneous space. It begins with a description of
the components, fundamental group, and cohomology of such configuration spaces
together with the physical interpretations of these results. The topological
arguments given generalize to Sobolev maps. The advantages of representing
homogeneous space valued maps by flat connections are described, with
applications to the homotopy theory of Sobolev maps, and minimization problems
for the Skyrme and Faddeev functionals. The paper concludes with some
speculation about the possiblility of using these techniques to define new
invariants of manifolds.
