Summary: In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups.