The special stability displayed by certain nuclei, the so called closed shell nuclei, can be understood in terms of the filling of particular orbitais of the mean field where the nucleons move essentially independently of each other. For nuclei with a number of nucleons outside closed shell, or a number of holes in the closed shell, the absolute minimum may correspond to a deformed configuration. The decay of the vibration into single-particle motion, known in the damping of plasmons in infinite systems as Landau damping, plays little role in the nuclear case, exception made in light nuclei. The Random Phase Approximation provides a diagonalization of the particle-vibration coupling Hamiltonian within the harmonic approximation. The basic importance of sum rules in the study of vibrational motion is that they are connected with basic operator identities which restrict the possible matrix elements in a physical system.