  We review an extension of Migdal's Theory of Finite Fermi Systems which has
been developed and applied to collective vibrations in closed shell nuclei in
the past ten years. This microscopic approach is based on a consistent use of
the Green function method. Here one considers in a consistent way more complex
1p1h$\otimes$phonon configurations beyond the RPA correlations. Moreover, these
configurations are not only included in the excited states but also explicitly
in the ground states of nuclei. The method has been applied to the calculation
of the strength distribution and transition densities of giant electric and
magnetic resonances in stable and unstable magic nuclei. Using these
microscopic transition densities, cross sections for inelastic electron and
alpha scattering have been calculated and compared with the available
experimental data. The method also allows one to extract in a consistent way
the magnitude of the strength of the various multipoles in the energy regions
in which several multipoles overlap. We compare the microscopic transition
densities, the strength distributions and the various multipole strengths with
their values extracted phenomenologically.
