We introduce the notion of a plabic tangle, which is the data of a planar bipartite graph drawn inside a disk called an "input disk," together with one or more "output" disks, which lie in faces of the graph. We show that under appropriate hypotheses, a plabic tangle gives rise to a "promotion map," which is a rational map from the Grassmannian of the input disk, to a product of Grassmannians associated to the output disks. We provide a number of examples in which these maps are compatible with the cluster algebra structure on the Grassmannian. Our motivating example is the case of "BCFW promotion," which we used to prove the cluster adjacency conjecture for the amplituhedron. This is based on joint work with Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.