Two years ago, here in Seattle, Santos communicated his negative solution of the Hirsch conjecture. (The conjecture was, "in any d-polytope with n facets, any two vertices are connected by a path of at most n-d edges".) However, the Polynomial Hirsch Conjecture ("...at most P(n,d) edges, for a polynomial P that does not depend on the polytope") is still wide open. Here we present some recent progress. We show that the Polynomial Hirsch Conjecture is true for flag complexes; and more generally, for all simplicial complexes whose Stanley-Reisner ideal is generated in bounded degree. In particular, we show that every flag manifold is Hirsch. This extends a classical result by Provan and Billera ("the barycentric subdivision of every shellable sphere is Hirsch"). This is joint work with Karim Adiprasito.