Observation of exact backscattering of X-rays and studies of its energy and angular dependences; test of the validity of the dynamical theory of diffraction in the extreme case of exact backscattering; backscattering high-energy-resolution monochromators; backscattering interferometers, in particular of the Fabry–Pérot interferometer type; and precise, up to 5·10—9 Å, measurements of crystal lattice parameters: these are central topics of the paper. Special attention is paid to the selection of crystals to be used as backscattering mirrors. Noncubic crystals like Al2O3, SiC, etc., allow backscattering for X-rays with practically any energy above 10 keV. Feasibility of backscattering mirrors for Mössbauer radiation of 57Fe (14.4 keV), 151Eu (21.5 keV), 119Sn (23.9 keV), and 161Dy (25.6 keV) nuclei is demonstrated by Al2O3 crystals. A concrete design of a sapphire Fabry–Pérot–Bragg étalon is presented. © 1999, Kluwer Academic Publishers.