In this talk, based on joint work with Gonzalo Contreras, I will sketch a proof of the following theorem: on any closed manifold of dimension at least two with non-zero first Betti number, a $C^\infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. I will derive this result as a consequence of the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^\infty$-close Riemannian metric.