Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study the combinatorics of Gr"obner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of Sturmfels and Zelevinsky. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Pl"ucker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gr"obner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields, permutations, and semi-standard tableaux.