All connected finite dimensional spaces fall into three categories, for each of which we can give an explicit description of the behavior of the higher homotopy groups. This in turn provides good information about the rational homology of the loop space. An old conjecture asserts that the rational homology of the free loop space should show similar behavior which, by a theorem of Gromov would give lower bounds on the number of closed geodesics in a generic Riemannian manifold as a function of their length. I will also report some progress on this problem.