Very little is known about cosmology away from the homogeneous limit. In this talk I will describe a set of results pertaining to universes with arbitrarily large inhomogeneity and anisotropy. I will prove rigorously that the spatial topology strongly constrains the ultimate fate of the universe - for "most" spatial topologies and with matter satisfying the weak energy condition, the universe must expand forever at least somewhere, despite the formation of black holes and other strong gravitational effects. Furthermore, in the presence of a positive cosmological constant (or suitable inflationary potential) there must always be part of the universe that expands at least as fast as de Sitter spacetime. I will describe the implications of this result for the so-called "initial conditions problem" of early-universe inflation.