K\"ahler-Einstein metrics: Old and New

We present classical and recent results on K"ahler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course “K"ahler-Einstein metrics” given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.