We begin with a note of general character. Most readers are familiar with the theory of algebraic varieties over a closed field and have become accustomed to identifying a variety with the set of its geometric points. This gives only a superficial representation of the heart of problems in the nonclosed case. Therefore, in the studies of algebraic groups defined over a nonclosed field or ring, there is a strict necessity to consider these groups as group objects in the category of schemes. Although the theory of group schemes is explicitly presented in proceedings of the seminar of Grothendieck and Demazure [58], there exists a compromise way. Let k be an arbitrary field, ks be the separable closure of k, G = Gal(ks/k) be the Galois group of the extension ks/k, and X be an algebraic variety over the field k. Then we have an object containing abundant information about the variety X. This is the ks-variety X = X ⊗k ks, considered together with the action of the Galois group G on X by the second factor. Therefore, we obtain an abundant family of associated G-sets and Galois G-modules: points X(ks), differential forms, modules Pic X, Br X, and so on. Therefore, we can decompose the study of the variety X into two steps: the geometric step (the study of the variety X over a closed field ks), and the algebraic step (the study of Galois modules corresponding to the scheme X considered over the field k). It is impossible to include both of these steps in one work, since such a work would be too large. In this paper, we present a detailed review of the second, algebraic aspect of studying varieties; the geometric part is assumed to be known. Of course, the decomposition of the study into two steps is too rough. Even from the formal standpoint, the variety X is only one fiber of a k-scheme X. For example, in the case of number fields, there exist p-adic varieties X⊗kkp with their own families of Galois modules. Further, arithmetical questions demand that one define integral structures on a variety X. The following problems immediately arise: the classification of these integral structures, the calculation of reduction of the integral model, and so on. Tamagawa measures or the Siegel formula appeared as tools in this field. We recall some recent events. In his well-known report at the International Mathematical Congress (1962), A. Borel [3] presented many achievements and newly arisen problems in the arithmetic of linear algebraic groups, which was a new topic at that time. In particular, he pointed to the dramatic situation in the study of such important characteristics of connected algebraic k-groups G as the Tamagawa numbers τ(G) or the sets Ш(G) of principal homogeneous spaces of G, which are trivial in all completions of the number field k. Counter-examples to all hypotheses on the structure of the numbers τ(G) or [Ш(G)] were soon constructed. Now we see that, at that time, the arithmetic of algebraic groups did not have enough additional deep information on the birational essence of the variety of the group G. A very important source of additional information was the Hironaka paper [59] on the reduction of singularities. In their earlier studies in surface theory, Yu. I. Manin and I. R. Shafarevich called attention to the fact that the group H1(k,Pic X) is a birational invariant in the class of smooth projective surfaces X defined over a field k and successfully used this fact (Manin [30, 31]). The structure of the module Pic X gives information about the arithmetic of the surface X; this fact is noted by Segre [70]. Even the first application of birational tools to study the properties of linear groups gave nontrivial results (Voskresenskii [6]). The birational geometry gave new possibilities for the study of cause–effect relations in the vast world which is called the theory of linear algebraic groups and their homogeneous spaces. At present, the birational geometry of algebraic groups has its own individuality and contains a number of