Cluster algebras are commutative rings defined by a set of generators and relations and equipped with a rich combinatorial structure.   It turns out that coordinate rings of many important varieties from Lie theory are cluster algebras.   In this talk, we will discuss cluster structures in open Schubert varieties of the Grassmannian and their categorification via representation theory of preprojective algebras.  In particular, we will relate combinatorics of Postnikov’s plabic graphs and recent work of Leclerc on cluster structures in flag varieties. This naturally generalizes the known results for the Grassmannian, and has been conjectured for some time.   We will also examine new connections between Grassmannians and certain combinatorial objects called $sl_k$ friezes.