Postnikov defined the totally nonnegative Grassmannian as the part of the
Grassmannian where all Plücker coordinates are nonnegative. This space can
be described by the combinatorics of planar bipartite graphs in a disk, by
affine Bruhat order, and by a host of other combinatorial objects. In this
talk, I will recall some of this story, then talk about in progress joint work,
together with Chris Fraser and Jacob Matherne, which hopes to extend this
combinatorial description to more general partial flag varieties.