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 first present a characterization of the known Einstein metrics on del Pezzo surfaces which I proved several years ago, I will  then describe some recent results which considerably improve our  understanding of Einstein metrics on these 4-manifolds, and suggest new avenues of research.