  We give an introduction to the ideas behind G. S. Tseytin's 1958 construction
of a seven-relation semigroup with undecidable word problem. We give a history
of the ideas leading up to its construction, some intuition for the proof, and
provide an overview of some subsequent results and developments which stem from
this remarkable semigroup. An English translation of Tseytin's article by the
author of the present article is supplemented at the end of the article (the
Russian original was published in Trudy Mat. Inst. Steklov 52 (1958), pp.
172--189).
