A review is presented of theoretical and experimental studies that have established the possible existence of bound states of an electron or an exciton with an optical phonon (in semiconductors and ionic crystals). These states arise near a threshold, above which emission of an optical phonon is possible. The distinguishing features of bound states in which optical phonons participate is that they are nonconserved particles. The stability of the bound states is therefore due to the fact that their decay is forbidden by the energy and momentum conservation laws. Unlike virtual phonons of the polaron ``jacket'', the phonon in a bound state is ``almost real''. The onset of bound states is aided by strong electron-phonon interactions, by a strong magnetic field, by the large mass of the particle that is bound to the phonon, and by the small dispersion of the phonons. A position intermediate between bound states and ordinary polarons is occupied by hybrid states that are produced when the phonon energy coincides with one of the transition energies in the phononless system; in these states, the difference between the ``real'' and ``virtual'' phonons is lost. The existence of bound and hybrid states becomes manifest in a number of physical phenomena, primarily in optical effects.