We study the asymptotic behaviour of the volume growth function of simply connected, Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice Γ. If X is asymptotically 1/4-pinched, we prove that Γ is divergent, with finite Bowen-Margulis measure, and that the volume growth of balls B(x, R) in X is asymptotically equivalent to a purely exponential function c(x)e ω(X)R , where ω(X) is the volume entropy of X. This generalizes Margulis' celebrated theorem for negatively curved spaces with compact quotients. A crucial step for this is a finite-volume version of the entropy-rigidity characterization of constant curvature spaces: any finite volume n-manifold with sectional curvature −b 2 ≤ k(X) ≤ −1 and volume entropy equal to (n − 1) is hyperbolic. In contrast, we show that for spaces admitting lattices which are not 1/4-pinched, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen-Margulis measure, the growth function can be exponential, lower-exponential or even upper-exponential.