I will begin by explaining what Kähler-Einstein metrics are and why one might care about them. In particular, I will show how to reduce the Kähler-Einstein equations to a scalar elliptic PDE - the complex Monge-Ampère equation. An interesting class of model solutions is given by complex hyperbolic space and its quotients, in particular, those with finite volume cusps. The model cuspidal solution comes in a 1-parameter family, any two members of which are asymptotic to each other at an exponential rate. I will explain some recent joint work with Xin Fu (UC Irvine) and Xumin Jiang (Fordham) where we prove that any complete solution to the Kähler-Einstein equations on the same underlying complex manifold must decay to a specific member of this family at a doubly exponential rate.