  We prove the following entropy-rigidity result in finite volume: if $X$ is a
negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then
$Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$
has the same length spectrum of a hyperbolic manifold $X_0$, the it is
isometric to $X_0$ (we also give a direct, entropy-free proof of this fact). We
compare with the classical theorems holding in the compact case, pointing out
the main difficulties to extend them to finite volume manifolds.
