Abstract: Riemannian manifolds in which every geodesic is closed, have been studied since the beginning of last century, when Zoll showed the existence of a non-round metric on the 2-sphere all of whose geodesics are closed. Among the many open problems on the subject, a conjecture of Berger states that for any simply connected manifold all of whose geodesics are closed, the geodesics must have the same length. The result was proved in the case of the 2-sphere by Grove and Gromoll. In this talk, I will show recent work with B. Wilking, where the Berger conjecture is proved for spheres of dimension >3.