Let Q(X) be the vector space of holomorphic quadratic differentials on a Riemann surface X of genus greater than one. This parametrizes the following two spaces of geometric objects on the underlying topological surface S: First, by a theorem of Wolf, and independently Hitchin, Q(X) provides a parametrization of marked hyperbolic structures, namely the Teichmuller space of S. Second, by a theorem of Hubbard and Masur, there is a bijective correspondence between Q(X) and measured foliations on S. In this talk I shall describe generalisations of these results to the case when Q(X) is replaced by the space of meromorphic quadratic differentials with poles of higher order. The proofs involve harmonic maps of infinite energy. Part of this is joint work with Michael Wolf.