Gravitational instantons are some of the simplest Riemannian manifolds with special holonomy: they are noncompact, Ricci flat, hyperkähler four-manifolds. There is now a scheme for classifying all of them which is well underway. Many of these spaces arise as moduli spaces of solutions to certain gauge-theoretic equations. A proposal known as Boalch’s modularity conjecture suggests that they all arise in this way.
I will describe this area and set of problems in general terms, and conclude by mentioning some recent progress on this by various different groups of researchers.