The Yang-Baxter equation first arose in the context of theoretical physics and statistical mechanics.

In 1990 Drinfeld proposed studying so-called set theoretical solutions of this equation: such a solution is a set X together with a map r : X x X –> X such that (r x id)(id × r)(r × id) = (id × r)(r × id)(id × r). In 2007 Rump introduced a new algebraic structure called a brace for the purpose of studying set theoretical solutions of the Yang-Baxter equation, and in 2016 Bachiller observed that there is a connection between the classification of braces of order n and the classification of Hopf-Galois structures admitted by a Galois extension of degree n. We shall describe this connection and illustrate how it has been beneficial for both brace theory and Hopf-Galois theory, including some recent work of the speaker and Koch