This is a survey of the theory of birationally rigid Fano varieties. In the Introduction the notions of nonrationality, maximal singularity, and birational rigidity are discussed. Part 1 contains the following preliminaries: minimal model programme; log minimal model programme; movable log pairs; Noether–Fano–Iskovskikh inequality; cubic surfaces; Sarkisov programme; log adjunction and connectedness. In Part 2 the discussion of threefolds includes: quartic threefolds; sextic double solid; double cover of a quadric; intersection of a quadric and a cubic; weighted hypersurfaces. In Part 3 the following topic on higher-dimensional varieties are discussed: hypersurfaces; complete intersections; double spaces; triple spaces; cyclic covers.