Abstract: One-relator groups are one of the oldest objects in combinatorial group theory, and the open problem of the word problem for one-relation monoids have mystified researchers since the 1950s. In recent year, one-relation inverse monoids have become of interest due to the undecidability they produce. The three objects -- one-relator groups, monoids, and inverse monoids -- are, in fact, highly interrelated, and there are many deep theorems connecting the three. I will give an overview of the theory of these three areas, how they link together, where modern research lives, and what might happen next. No particular background beyond group theory will be used.