As explained in my colloquium talk, Seiberg-Witten theory has much to say about the scalar curvature of 4-manifolds. However, the Seiberg-Witten equations also turn out to be directly sensitive to the self-dual part of the Weyl curvature. By making judicious use of this observation, one can obtain sharp estimates for invariants defined in terms of the Ricci curvature or the sectional curvature. As an application, I will derive some new results concerning Einstein metrics.