  We investigate the spectral distribution of the damped wave equation on a
compact Riemannian manifold, especially in the case of a metric of negative
curvature, for which the geodesic flow is Anosov. The main application is to
obtain conditions (in terms of the geodesic flow on $X$ and the damping
function) for which the energy of the waves decays exponentially fast, at least
for smooth enough initial data. We review various estimates for the high
frequency spectrum in terms of dynamically defined quantities, like the value
distribution of the time-averaged damping. We also present a new condition for
a spectral gap, depending on the set of minimally damped trajectories.
