On derivations in group algebras and other algebraic structures

The work is devoted to a survey of known results related to the study of derivations in group algebras, bimodules and other algebraic structures, as well as to various generalizations and variations of this problem. A review of results on derivations in $L_1(G)$ algebras, in von Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed, in particular, derivations in groups, $(\sigma,\tau)$-derivations, and the Fox calculus. A review of some results related to the application to pseudodifferential operators and the construction of the symbolic calculus is also given. In conclusion, some results related to the description of derivations as characters on the groupoid of the adjoint action are described. Some applications are also described: to coding theory, the theory of ends of metric spaces, and rough geometry.