Cluster algebras are a class of commutative rings with a special set of generators endowed with a nice combinatorial structure. A traditional problem in the field pertains to rings of regular functions over some algebraic variety: whether they possess a cluster algebra description, and if so, how to describe as many clusters as possible. Weaves are a type of diagram first discovered in connection to symplectic geometry. In this talk, we will describe braid varieties, a large class including (open) Richardson varieties; and we show how to use weaves to describe a large set of clusters in the coordinate rings of these varieties.