In 1980s, Yau conjectured that the existence of Kaehler Einstein metric on Fano manifold is related to an algebraic geometric condition of "stability"´. The recent work with Donaldson, Sun Song confirmed this conjecture. In the talk, we will review history of this problems as well as this subject, and we also will review earlier work of G. Tian and others on this problems. We will outline the strategy of proof, which involves deforming through metrics with cone singularities. If time permits, we will give more details about various aspects of the proof.

This talk is aimed at a general mathematical audience.