  We prove that a $C^2$-generic Riemannian metric on a closed surface has
either an elliptic closed geodesic or an Anosov geodesic flow. As a
consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic
flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow
of a closed surface is Anosov. In order to prove these statements, we establish
a general result that may be of independent interest and provides sufficient
conditions for a Reeb flow of a closed 3-manifold to be Anosov.
