In 2005, Kechris, Pestov and Todorcevic exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the minimal actions of some subgroups of $S_\infty$. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons in several directions.