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Cluster varieties were introduced by Fock and Goncharov as geometric enrichments of cluster algebras of Fomin and Zelevinsky. Since their inception, cluster variety structures have been observed in many geometric objects of a wide range of research interests and have provided new insights to many problems in different areas of mathematics. In this talk, we will focus on one family of examples called flag moduli spaces, which are configuration spaces of flags satisfying relative position conditions imposed by a positive braid. We will describe their cluster structures using Legendrian weaves and discuss how to use these cluster structures to solve the infinitely-many-filling problem of Legendrian links. In the end, we will also mention some open problems for future research.