I will discuss the connection between Sasaki-Einstein metrics, or conical Ricci flat Kahler metrics, and the algebro-geometric notion of K-stability. In particular, I will give a differential geometric perspective of K-stability which arises from the Sasakian view point, and use K-stability to find infinitely many non-isometric Sasaki-Einstein metrics on the 5-sphere. This is joint work with G. Szekelyhidi. Speaker(s): Tristan Collins (Harvard)