Abstract: Like hyperbolic space, every complete, simply connected Riemannian manifold M with uniformly negative curvature can be compactified in a canonical way by adding a sphere at infinity. This ideal boundary carries a kind of "generalized quasiconformal structure" which captures the asymptotics of M. It turns out that when dim(M) = 3, one can use the ideal boundary to recast Thurston's hyperbolization conjecture as the problem of finding a quasiconformal parametrization of the boundary by the standard sphere, i.e. as a kind of generalized uniformization problem. The lecture will discuss this circle of ideas, and relations with recent work involving analysis on metric spaces by Semmes, Heinonen, Koskela, and others.