The article reviews the current state of the study of the dielectric properties of a medium in the x-ray and $\gamma$-ray frequency regions. Particular attention is devoted to the systematic introduction of the dielectric permittivity tensor in the x-ray region, starting from the most fundamental property of crystals–spatial periodicity. The properties of the permittivity tensor are treated both in terms of the phenomenological theory and on the basis of specific microscopic models. The review includes theoretical and experimental investigations of the contributions of the electronic and nuclear subsystems to the permittivity of crystals. The polarizability of the electron subsystem is considered in the strong-coupling approximation and in terms of the Bloch formalism. Considerable space is devoted to discussion of the permittivity near isolated low-lying nuclear transitions in the presence of hyperfme interactions, where its properties are substantially more complicated and more interesting than those of the permittivity of the electron subsystem. A review is given of the principal phenomena of the polarization optics of Mossbauer radiation. The review analyzes the symmetry properties of the x-ray permittivity and establishes a number of useful relations regarding generalized symmetry operations.