Braid varieties are smooth affine algebraic varieties constructed from the data of an element in the positive braid monoid of a simple Lie algebra. They generalize prominent algebraic varieties in the context of Lie theory, such as Richardson varieties and double Bott–Samelson cells. Their coordinate algebra admits a locally acyclic cluster structure of really full rank, and many important cluster algebras (such as those of finite cluster type) can be realized as the coordinate ring of a braid variety, up to frozen variables. In the first lecture, I will define braid varieties and explain how other algebraic varieties appearing in Lie theory can be realized as braid varieties. In the second lecture, I will sketch the construction of a cluster structure on the coordinate algebra of a braid variety, and elaborate on basic properties of this structure. In the last lecture I will give further properties of the cluster structure, and some consequences this structure has on the geometry of the braid variety and other closely related varieties.