We consider some novel topological effects, with potentially observable consequences, in classical and quantum gravity. In one scenario, we find that the topology of extra dimensions generically breaks global Lorentz symmetry, leading to distinct experimental signatures in the context of brane worlds. In a different scenario, we show that topological terms in the gravitational action, such as those expected in heterotic string theory, can greatly enhance the instability of four-dimensional de Sitter space by favouring the nucleation of primordial black holes.