*Kähler-Einstein metrics*was initiated by E. Calabi in the 50's. In the 70's, Yau's celebrated solution of the Calabi conjecture yielded the existence of

*Kähler-Einstein metrics*with vanishing first Chern class. Aubin and Yau solved the existence problem for the case of negative first Chern class.

Since then, it has been a challenging problem to study the existence of *Kähler-Einstein metrics* on Fano manifolds. A Fano manifold is a compact Kähler manifold with positive first Chern class. There are obstructions to the existence of *Kähler-Einstein metrics* on Fano manifolds. Recently, the problem has been solved by relating existence to K-stability, a geometric stability generalizing substantially the stability from classical geometric invariant theory.

In the first lecture, I will first recall some known facts about Riemann surfaces. Then we give a brief tour on the study of *Kähler-Einstein metrics* in the last two decades. Next I will explain how those Einstein metrics are related to the study of algebraic group actions and geometric stability.

This lecture is aimed at a general audience.