The ends of an infinite-volume hyperbolic 3-manifold have a rich and mysterious geometric structure, which has been studied using methods of complex analysis, dynamics, topology and geometry. Thurston conjectured in the 1980's that this structure is completely classified by "end invariants" which describe its asymptotic properties. Recently in joint work with J. Brock and R. Canary we were able to prove this conjecture (in the incompressible-boundary case), using in an essential way the combinatorial structure of the set of closed curves on a surface. I will give an overview of the structure of this field and of these and related developments.