This report is devoted to the studies of multiphonon excitations in nuclei, it contains in fact two reports: an experimental one presenting a review of the observation of multiphonon excitations and a theoretical one centered on the description of such states. The present knowledge about giant resonances in nuclei is first briefly presented and the relative merits of different probes to excite such states are illustrated. The existence of giant resonances built on excited states is stressed. In this context, multiphonon states, i.e. a giant resonance built on top of other giant resonances, are expected. Several paths can be followed to obtain experimental evidence for multiphonon states: inelastic nuclear and Coulomb excitations and charge exchange reactions. The status of the search for isoscalar multiphonon excitations by means of the strong nuclear potential produced by heavy ions is presented and novel experimental signatures of the double excitation of the giant quadrupole resonance are reported. Coulomb excitation induced by relativistic heavy ions is presented as another fruitful approach to the study of the double dipole and recent experimental results using the very selective ( π+, π-) double charge exchange reactions are shown and discussed. Calculations for these different excitation mechanisms are presented. A review of the theoretical descriptions of the properties of the multiphonon states is then presented. Hydrodynamical and phenomenological models are first discussed. The mean-field approaches are summarized. The random phase approximation is introduced to discuss giant resonances while several extensions of the meanfield concepts are presented in order to deal with large amplitude motion and multiphonons. Finally, boson mapping techniques are used to introduce a general treatment of multiple excitations. The different approaches consistently predict that such multiple collective excitations in nuclei should closely follow a harmonic pattern. Therefore, the energy of the phonons is predicted to be additive. This conclusion allows to discuss the properties of the width of the multiphonon strength in a simple manner and the observed width is predicted to be close to the quadratic sum of the single-phonon widths. General conclusions are drawn and new prospects are discussed.