Sasakian Geometry and Einstein Metrics on Spheres

This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Math'ematiques, Universit'e de Montr'eal, during the period June 28-July 16, 2004. It is a report on our joint work with J'anos Koll'ar concerning the existence of an abundance of Einstein metrics on odd dimensional spheres, including exotic spheres.