This talk focuses on the recent resolution of the following conjectures in the study of Ricci-flat four manifolds (joint work with Song Sun).

(1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.

(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have one of the following asymptotic model geometries: ALE, ALF, ALG, ALH, ALG* and ALH*.

We will also discuss some applications and further directions.