Cluster algebras, defined by Fomin and Zelevinsky around the year 2000, are a special kind of commutative algebras with amazing combinatorial properties. Since their inception, they have been connected to mirror symmetry, total positivity, Lie theory and link invariants, among others. Unlike the commutative algebras we encounter early-on, cluster algebras are not usually given by generators and relations, which makes it hard to tell whether a given algebra has a cluster structure. I will describe a construction of cluster structures on coordinate rings of varieties defined in terms of positions of flags. This construction unifies a large part of the motivating examples of the theory of cluster algebras and also includes new, long-conjectured cluster structures on Lie-theoretic varieties. The construction makes critical use of combinatorial gadgets called weaves, which are heavily inspired by recent constructions in symplectic and contact geometry. The talk will touch upon joint works with various subsets of {Marco Castronovo, Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, Linhui Shen, David Speyer}.