The space of Kahler-Einstein (KE) Fano manifolds can be naturally compactified by considering metric degenerations in the so-called Gromov-Hausdorff topology. The relation between the existence of a KE metric and an algebro-geometric notion of stability of the underlying variety (Yau-Tian-Donaldson Conjecture, now a theorem), suggests that the above metric compactification should correspond to certain compact algebraic moduli spaces of stable Fano varieties (K-Moduli).
In this talk we will describe the general picture, focusing on the understood case of Del Pezzo surfaces (joint work with Yuji Odaka and Song Sun).