Quasisconformal homeomorphisms -- a class of homeomorphisms generalizing conformal diffeomorphisms -- have served as a powerful tool in Teichmuller theory, complex dynamics, the theory of Kleinian groups, and in rigidity theorems like Mostow rigidity. In fact, the definition makes sense in the setting of metric spaces, and it has now become clear that a suitable extension of the classical theory to more general metric spaces would have important applications to the theory of negatively curved manifolds, and to Thurston's Hyperbolization conjecture for 3-manifolds. In the lecture I will explain the background material, and discuss recent joint work with Mario Bonk.