Abstract: We discuss some global properties of asymptotically de Sitter spacetimes satisfying the Einstein equations with positive cosmological constant. Interest in such spacetimes has increased in recent years due to continuing evidence supporting an accelerating cosmos. The results we present, although purely classical, were originally motivated by work of Witten and Yau on the AdS/CFT correspondence, which illustrates how the geometry and/or topology of the conformal infinity (in the sense of Penrose) of an asymptotically hyperbolic Riemannian manifold can influence the global structure of the manifold. In this vein, the results we present establish connections between the geometry/topology of conformal infinity and the occurrence of singularities in asymptotically de Sitter spacetimes. For example, it is shown, under appropriate energy conditions, etc., that if future conformal infinity is of negative Yamabe class then all timelike geodesics in spacetime must be past incomplete. This and other related results are illustrated by well-known examples. This talk is based on joint work with Lars Andersson.