Kuperberg's webs give a diagrammatic calculus for spaces of tensor invariants of quantum groups. These are best understood in the cases of U_q(sl_2) and U_q(sl_3), when it is possible to choose a subset of the webs forming a distinguished basis (the non-crossing matchings for sl_2 and "non-elliptic" webs for sl_3). These were used by Khovanov in his categorification of the corresponding quantum link invariants. I will describe an ongoing research program, joint with Pechenik, Pfannerer, Striker, and Swanson, in which we interpret webs as generalizations of Postnikov's plabic graphs. This perspective allows for a unified combinatorial description of the known distinguished web bases and the discovery of the first such basis for sl_4.