Consider a compact $n$-manifold $M$ of Ricci curvature bounded below,normalized by $(n-1)H$,where $H=\pm 1$ or $0$.Then $M$ is isometric to a $H$-space form under either of the maximal volume conditions:(i)(Bishop)there is $\rho>0$ such that every $\rho$-ball on $M$ has the maximal volume i.e.,the volume of a $\rho$-ball in a simply connected $H$-space form;(ii)(Ledrappier-Wang)For $H=-1$,the volume entropy of $M$ is maximal i.e.,the volume entropy of any hyperbolic$n$-manifold.