A rewriting system comprises an alphabet and some rules for simplifying words over the alphabet.  Each rewriting system ``presents'' a monoid, and sometimes that monoid is a group.  It is natural to ask which groups admit a presentation by particularly nice rewriting systems.  In 1984 Gilman conjectured that the groups which can be presented by finite convergent monadic rewriting systems are exactly the plain groups.  We will discuss a proof, discovered in collaboration with Andy Eisenberg (St Louis University), that Gilman was right.  We will also discuss questions concerning groups presented by length-reducing rewriting systems which remain unresolved.