Generalized compactifications of string theory relax the condition that there be a globally well defined space in the extra dimensions. We give up the usual distinction between a Riemannian manifold and other fields living on that manifold, so that the two become intrinsically intertwined. Such compactifications are the natural result of lifting generic gauging of lower dimensional supergravity theories to the full 10D string theory. But, what is the topological choice that *defines* such a string theory compactification? In my talk, I will advocate one possible answer: the "doubled geometry" seen simultaneously by left plus right moving string excitations. This geometry neatly packages the fields into purely metric degrees of freedom. Its topology unifies the traditional choice of spatial topology and magnetic flux into a single generalized choice. I'll begin with a simple motivating example, then describe the framework of Hull and Reid-Edwards for locally reconstructing physical from doubled geometry. Finally, I'll describe insights from applying this framework to the SU(2) WZW model, a simple soluble model which nevertheless exhibits interesting and surprising features.