In 1914, A. Thue (the PhD supervisor of T. Skolem) posed an innocent-looking problem about transforming some words into others, subject to a fixed set of rules. This problem -- the word problem for finitely presented semigroups -- would come to have a remarkable effect on the development of mathematical logic, group theory, and semigroup theory in the half century to come. Indeed, it can be seen as one of the key links ensuring that mathematical logic became a firm part of mathematics in the 1930s and 1940s. In this talk, I'll give an overview of the problem, its history, and how it developed. I will present a special case of it -- the word problem for one-relation semigroups -- which despite an inordinate amount of effort to crack, remains an unsolved problem. Finally, I will present some of my own efforts to approach and understand this wonderful problem.