Geometers are interested in the problem of finding a "best"
metric on a manifold. In dimension 2, the best metric is usually one
which possesses the most symmetries, such as the round metric on a
sphere, or a flat metric on a torus. In higher dimensions, there are
many more classes of geometrically interesting metrics. I will give a
general overview of a certain class of Einstein metrics in dimension 4
which have special holonomy, and which are known as "gravitational
instantons." I will then discuss certain aspects of their classification
and connections with algebraic surfaces.