In this thesis we study the vectorial damped wave equation on a compact Riemannian manifold with no boundary. The damping term is thus a matrix valued function and the solutions of the equation are vector valued.We first study the problem of stabilization and compute the best exponential decay rate of energy of the solutions of the equation. This result allows us to prove a necessary and sufficient condition for strong stabilization. We also exhibit the apparition of a new phenomenon of high frequency overdamping.We then study the asymptotic repartiton of the eigenfrequencies. We show that, up to a null density subset, all the eigenfrequencies are concentrated in a strip parallel to the imaginary axis. The width of this strip is determined by the Lyapunov exponents of some dynamical system constructed from the damping term of the equation.