The paper under review is a survey devoted to main results on elementary properties of groups with the main focus on linear groups. Section 1 contains necessary information about model theory. Results presented in Section 2 are devoted to the classification of groups up to their elementary properties. Section 3 contains results on elementary classification of classical linear groups. Section 4 is devoted to other results on the elementary equivalence of groups and related structures, in particular, of the lattices of subalgebras of free algebras of varieties, of modules and their tensor products, free products of groups, filtered and wreath products. Section 5 is devoted to categoricity and stability. In particular it contains results on \(\varpi\)-categorical, \(\varpi\)-stable and \(\varpi_1\)-categorical groups. Section 6 contains results on decidable and undecidable group theories.