The paper under review provides a vast account for the recent effective results on birational (super)rigidity of algebraic varieties. The first chapter studies birational superrigidity of index \(1\) Fano hypersurfaces with quadratic singularities (see \S 1 of the paper) and, more generally, of index \(1\) Fano complete intersection with multiquadratic singularities (see \S 3). Assuming that any such variety \(X\) also satisfies certain generality condition (cf. Definition 2.1 in the text), the author establishes the \textit{generalized \(4n^2\)-inequality} (see Theorem 2.1), from which birational superrigidity of \(X\) basically follows. In addition the codimension of those \(X\) that are \textit{not} birationally superrigid is estimated from below (see Theorems 1.2 and 3.3). All this is a recap of results in [\textit{T. Eckl} and \textit{A. Pukhlikov}, Springer Proc. Math. Stat. 79, 121--139 (2014; Zbl 1327.14191)] and in [\textit{D. Evans} and \textit{A. V. Pukhlikov}, Izv. Math. 83, No. 4, 743--769 (2019; Zbl 1427.14033); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 83, No. 4, 100--128 (2019)]. The second chapter discusses Fano-Mori fibre spaces \(\pi: V \longrightarrow S\) (here \(V\) is a rationally connected variety and \(\dim V - \dim S \ge 3\)). Under certain conditions (see (i)--(iii) in Theorem 1.1), including \(\text{lct}(F_s) = 1\) for the \textit{global log canonical threshold} of every fiber \(\pi^{-1}(s)\), the \textit{\(K\)-condition} on the linear systems \(|-nK_V + \pi^*Y|\) and so on, the author proves that \(V \slash S\) is birationally superrigid (the existence of \(V\) satisfying the given conditions is provided by Example 1.2). The argument here is mainly based on the construction of a \textit{mobile family} \(\mathcal{C}\) of irreducible rational curves on \(V\) (intersecting with general \(C \in \mathcal{C}\) one excludes the linear systems coming from birational maps \(V \slash S \dashrightarrow V' \slash S'\) to other Fano-Mori fibre spaces). A detailed proof is contained in [\textit{A. V. Pukhlikov}, Izv. Math. 86, No. 2, 334--411 (2022; Zbl 1502.14035); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 86, No. 2, 128--212 (2022)]. The chapter closes with a discussion of the estimate \(\mathrm{ct} \ge 1\) for the canonical threshold of Fano hypersurfaces considered in Chapter 1 (see \S 2) and how the method of maximal singularities extends to study the birational geometry of Fano varieties having index \(\ge 2\) (see \S 3).