Kohnert gave an elegant combinatorial rule for computing Demazure characters for the general linear group via pushing down cells on left-justified diagrams of cells in the first quadrant of the plane. He conjectured that Schubert polynomials could be computed in the same way beginning with Rothe diagrams for permutations, and several (some contentious) proofs of this appear in the literature. In joint work with Dominic Searles, we consider Kohnert’s algorithm applied to any diagram and study the resulting polynomials, which we call Kohnert polynomials. These simultaneously generalize both Schubert polynomials and Demazure characters, and in general have nice combinatorial and algebraic properties. After surveying properties of general Kohnert polynomials, I’ll focus on two new Kohnert bases, one of which is conjecturally Schubert positive and stabilizes to skew Schur functions, and the other stabilizes to a new basis of quasi symmetric functions that contains the Schur functions.