Let \(M\) be a compact differentiable manifold of dimension \(n \ge 3\) endowed with a Riemannian metric or a reversible Finsler metric (in the second case, the norm \(\|v\|\) of a tangent vector \(v\) is a positive definite function \(f(v)\), the term reversible means that \(f(-v)=f(v)\) for all vectors \(v\)). The main result of the paper is the following: For a \(C^r\)-generic Riemannian metric with \(r \ge 2\) or a \(C^r\)-generic reversible Finsler metric with \(r \ge 4\), all prime closed geodesics are simple, and geometrically distinct closed geodesics do not intersect each other. Moreover, for a generic Riemannian metric on a compact and simply connected manifold \(M\) all closed geodesics are simple and the number \(N(t)\) of geometrically distinct closed geodesics of length \(\le t\) grows exponentially as \(t \to \infty\).