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. Tea at 3:30 pm in SC 1425.