Birationally rigid Fano varieties

The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample to the three-dimensional Lüroth problem. Since then, birational rigidity and superrigidity have been proved for a broad class of higher-dimensional varieties, among which the Fano varieties occupy the central place. The present paper is a survey of the theory of birationally rigid Fano varieties.