Combinatorial methods: from groups to polynomial algebras

Combinatorial methods (or methods of elementary transformations) came to group theory from low-dimensional topology in the beginning of the century. Soon after that, combinatorial group theory became an independent area with its own powerful techniques. On the other hand, combinatorial commutative algebra emerged in the sixties, after Buchberger introduced what is now known as Gr\"{o}bner bases. The purpose of this survey is to show how ideas from one of those areas contribute to the other.