On the number of closed geodesics

It is an open question whether any Riemannian metric on a compact manifold has infinitely many geometrically distinct closed geodesics. We first revisit results for generic metrics in the case of simply-connected manifolds. The genericity assumption is defined in terms of the linearization of the local return map of the periodic orbits of the geodesic flow. Then we discuss recent results obtained jointly with Iskander Taimanov on the number of closed geodesics on connected sums of non-simply connected and compact manifolds. In particular we conclude that any Riemannian metric on a three-dimensional compact manifold with infinite fundamental group has infinitely many closed geodesics.