`Quasi-mosaicity' is an effect of anisotropy in crystals that permits one to obtain a curvature of internal crystallographic planes that would be flat otherwise. The term `quasi-mosaicity' was introduced by O. Sumbaev in 1957. The concept of `quasi-mosaicity' was then retrieved about ten years ago and was applied to steering of charged-particle beams at the Super Proton Synchrotron at CERN. Beams were deviated by exploiting channeling and volume reflection phenomena in curved crystals that show the `quasi-mosaic' effect. More recently, a crystal of this kind was installed in the Large Hadron Collider at CERN for beam collimation by the UA9 collaboration. Since 2011, another important application involving the `quasi-mosaic' effect has been the focalization of hard X-rays and soft γ-rays. In particular, the possibility of obtaining both high diffraction efficiency and the focalization of a diffracted beam has been proved, which cannot be obtained using traditional diffracting crystals. A comprehensive survey of the physical properties of `quasi-mosaicity' is reported here. Finally, experimental demonstrations for adjustable values of the `quasi-mosaic' curvature are provided.