Simple closed geodesics in dimensions $\ge 3$

We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold $M$ of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras~\cite{C2010} \cite{C2011} this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number $N(t)$ of geometrically distinct closed geodesics of length $\le t$ grows exponentially.