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.