This review discusses the properties of highly excited vibrational states of polyatomic molecules and molecular crystals. As we know, one can describe small vibrations of molecules with the concept of normal modes. In molecules having several identical valence bonds (C$_6$H$_6$, H$_2$O, etc.) the normal modes that describe the vibrational excitations of these bonds amount to vibrations whose energy is more or less uniformly distributed over all the bonds, with a degree of delocalization of the energy over the bonds that increases with increasing level of excitation. On the other hand, an extensive set of physical phenomena exists (e.g., dissociation of molecules) in which local excitations, a considerable fraction of which are spatially localized, play an important role. A localized state corresponds to a complicated superposition of normal modes. Hence the concept of normal vibrations is inadequate for describing vibrations (or, better expressed, movements) of a highly excited molecule. One can conveniently describe such movements of the molecule in the representation of local modes (LM). As a rule, one takes an LM to mean simply the coordinate of a valence bond of the molecule, e. g., O–H, C–H, etc. A large number of papers has been published recently on the experimental study of LM and infrared spectra, relaxation experiments, selective photochemistry, etc. This review casts light on these experimental data on the basis of the theory of LM.