A theme of long standing interest (to the speaker!) concerns the relationship between the topology of spacetime and the occurrence of singularities (causal geodesic incompleteness). Many results concerning this center around the notion of topological censorship, which has to do with the idea that the region outside all black holes (and white holes) should be topologically simple. In this talk we present results which provide support for topological censorship at the pure initial data level, thereby circumventing difficult issues of global evolution. The proofs rely on the recently developed theory of marginally outer trapped surfaces, which have played an important role in the theory of black holes, and which may be viewed as spacetime analogues of minimal surfaces in Riemannian geometry. The talk will begin with a brief overview of general relativity and topological censorship. The talk is based primarily on joint work with various collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair and Dan Pollack.