Abstract: I will introduce and discuss a family of affine algebraic varieties, called braid varieties. Such a variety is associated with a simply-connected simple algebraic group G and a positive braid of the corresponding type. Braid varieties generalise various nice subvarieties of flag varieties and Grassmannians, and are closely related to character varieties and to augmentation varieties for Legendrian links. Their topology is related to link invariants such as Khovanov-Rozansky homology, while their rings of regular functions carry cluster algebra structures. The talk is based on a series of joint works with Roger Casals, Pavel Galashin, Eugene Gorsky, Ian Le, Linhui Shen, Melissa Sherman-Bennett, and José Simental.