There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there might also exist other Einstein metrics on them that are utterly different from the ones we currently know.

In this lecture, I will present two quite different characterizations of the known Einstein metrics on del Pezzo surfaces. These results imply, in particular, that the known Einstein metrics exactly sweep out a single connected component of the Einstein moduli space. I will then briefly indicate the role these results play in current avenues of research.