  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.
