Abstract: Let M be a closed Riemannian manifold. The volume entropy h(M) of M is defined to be the asymptotic exponential growth rate of the volume of balls centered at a fixed point tilde p in the universal cover tilde M as the radius goes to infinity. If Ricci curvature of M >= -(n-1), then h(M)<=(n-1) and "=" holds if and only if M is isometric to a hyperbolic manifold (Ledrappier-Wang). We prove that if h(M)>(n-1-epsilon), where epslion depends only on the diameter and the dimension n, then M is Gromov-Hausdorff close and diffeomorphic to a hyperbolic manifold. By Besson-Courtois-Gallot's minimal volume theorem, M share the same volume lower bound v(n)>0 of hyperbolic manifolds. We also prove that almost maximal volume entropy is equivalent to local maximal volume on the universal cover. Similar diffeo/isom rigidity holds for positive/nonnegative lower Ricci curvature bound. It is a joint work with Lina Chen and Xiaochun Rong.