Abstract: One of the major themes of modern differential geometry is the relationship between the curvature and topology of Riemannian manifolds. In this lecture, I will describe some links between curvature and SMOOTH topological invariants --- i.e. invariants which can distinguish between different smooth structures on a given topological manifold. The specific invariants I will discuss are the Seiberg-Witten invariants of 4-manifolds, and I will describe the impact these have on the existence problem for Einstein metrics and some related Riemannian variational problems.