Gróbner–Shirshov bases: from incipiency to nowdays

In this paper we present a history and the main results of the Gróbner–Shirshov bases theory for commutative, non-commutative, Lie and conformal algebras from the beginning (1962) up to nowadays. We consider the problem of constructing free Lie algebra basis, structure of free products of Lie algebras, word problem for Lie algebras, embedding of arbitrary Lie algebra into algebraically closed one. The contemporary form of Composition–Diamond lemma (CD-lemma) is adduced. The rewriting systems for groups are considered from the Gróbner–Shirshov bases theory point of view. The important role is devoted to conformal algebras, we present a statement of CD-lemma for associative conformal algebras, some examples are considered. The analogue of Hilbert basis theorem for commutative conformal algebras is stated.