Abstract: These metrics have been a central topic in complex differential geometry for the past half century. As the name suggests, they are related (at some distance) to general relativity, but also have fundamental connections with algebraic geometry. The first part of the talk will aim to give a glimpse of these general contexts. Then we discuss briefly some history of the work on the question of existence of Kahler-Einstein metrics, as developed by Calabi, Yau, Aubin, Tian and others. The last part of the talk will outline recent work of the speaker with Chen and Sun which establishes a long-conjectured relation between the existence of Kahler-Einstein metrics and an algebro-geometric notion of “stability”, which we will also attempt to explain.