The basic physical principles of resonance hyper-Raman scattering (RHRS) of light by media in different aggregate states are examined. The theory of RHRS by molecules, which takes into account the vibronic interaction between the resonance electronic states of a molecule, is presented. Expressions are given for the components of the tensor of RHRS in vibrational and vibrational-rotational transitions. The selection rules, the symmetry of the RHRS tensor, the polarization characteristics of the process, and the ratios of the intensities of scattering by totally and nontotally symmetric vibrations and their overtones are discussed. The results of experimental investigations of RHRS by molecules in the gas phase and molecules adsorbed on ultradispersed particles of silver are presented. These results confirm that the proposed theory adequately describes the actual process of RHRS qualitatively and in a number of cases quantitatively also, and they demonstrate that RHRS is a good source of spectral information. Data on RHRS by phonons in solids, for which there is still no theory, also show that RHRS spectroscopy could potentially yield a great deal of information.