I will describe some progress in determining the conditions for existence of canonical metrics in Kahler geometry. From the PDE point of view, it amounts to solving certain complex Monge-Ampere equations. Picking out a canonical metric in a Kahler class amounts to understanding the moduli space of such classes, and there is an obstruction to the existence of a smooth solution in terms of an algebraic stability condition. I will show how to tell whether a manifold is K-stable and identify destabilizing degenerations by taking a variational approach and computing the asymptotics of certain singular rational integrals.