We study the asymptotic behavior of simply connected Riemannian manifolds~$X$ of strictly negative curvature admitting a non-uniform lattice $\Gamma$. If the quotient manifold ${\bar X= \Gamma \backslash X}$ is  asymptotically $1/4$-pinched, we prove that  $\Gamma$  is divergent and $U\bar X$ has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show  that, in this case,  the volume growth of balls $B(x,R)$ in $X$ is asymptotically equivalent to a purely exponential function $c(x)e^{\delta R}$, where $\delta$ is the topological entropy of the geodesic flow of $\bar X$.   This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast,  we exhibit examples  of lattices $\Gamma$ in negatively curved spaces $X$ (not asymptotically $1/4$ pinched)    where, depending on the critical exponent of the parabolic subgroups   and on the finiteness of the Bowen-Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential.