For a finite extension of fields L/K, a Hopf-Galois structure on L/K is a K-Hopf algebra together with a certain action on L. Hopf-Galois structures on Galois extensions of number fields are a subject of great interest in Galois module theory as they reveal information about the rings of integers of these extensions. On the other hand, the Yang-Baxter equation is a matrix equation for the linear automorphisms of the tensor product of a vector space with itself. The Yang-Baxter equation is one of the important equations in quantum group theory, which has applications in mathematical physics. The classification of Hopf-Galois structures and solutions of the Yang-Baxter equation both remain among important topics of research. In this talk we will first explain how Hopf-Galois theory and the Yang-Baxter equation came to be related via algebraic objects known as skew braces. Then we will review our results on classification of Hopf-Galois structures and skew braces of order p³.