It is well-known that plabic (planar bi-colored) graphs, up to equivalence, define clusters for the cluster algebra of the Grassmannian. Moreover, square moves on plabic graphs correspond to mutations between clusters. However, not all seeds in a cluster algebra correspond to plabic graphs. I will explain how weaves, another planar combinatorial object, generalize plabic graphs. Weaves are robust enough to, conjecturally, give all clusters in the cluster structure for the Grassmannian.