I will explain both the original definition of a cluster algebra, and the recent recasting by Gross, Hacking, Keel and Kontsevich in terms of a holomorphic symplectic variety with a compatible dense torus. I will then present several of the most important examples of cluster varieties and see how their cluster structure illuminates their geometry.

In the second lecture, I will present work in progress which attempts to describe the (de Rham) cohomology of cluster varieties and the mixed Hodge structure on it. Neither background in cluster varieteis (beyond the first talk) nor in Hodge theory will be assumed. Joint work with Thomas Lam.