Associated to a positive braid, we define an affine algebraic variety via an explicit set of polynomial equations. I will give properties of these varieties, including their dimension, smoothness properties and a realization as a moduli space of chains of flags. I will also explain how some classical varieties in Lie theory, such as positroid and more generally Richardson varieties, appear in this way, as well as a connection to the computation of the Khovanov-Rozansky homology of the link obtained by closing the braid. This is joint work with Roger Casals, Eugene Gorsky and Mikhail Gorsky.