  This survey is intended to provide an overview of one of the oldest and most
celebrated open problems in combinatorial algebra: the word problem for
one-relation monoids. We provide a history of the problem starting in 1914, and
give a detailed overview of the proofs of central results, especially those due
to Adian and his student Oganesian. After showing how to reduce the problem to
the left cancellative case, the second half of the survey focuses on various
methods for solving partial cases in this family. We finish with some modern
and very recent results pertaining to this problem, including a link to the
Collatz conjecture. Along the way, we emphasise and address a number of
incorrect and inaccurate statements that have appeared in the literature over
the years. We also fill a gap in the proof of a theorem linking special inverse
monoids to one-relation monoids, and slightly strengthen the statement of this
theorem.
