Toric matroid bundles are combinatorial objects which result from tropicalizing vector bundles over toric varieties. I'll explain how to construct toric matroid bundles, how to check if a toric matroid bundle is globally generated or ample, how to compute the Euler characteristic, and how to compute the characteristic classes of a toric matroid bundle in the T-equivarient Chow cohomology of the base.

Finally, I'll explain how each matroid determines a tautological toric matroid bundle over the permutahedral toric variety. I'll discuss some properties of these bundles, and I'll show that the characteristic classes of the tautological toric matroid bundle recover the tautological classes of matroids used by Berget, Eur, Spink, and Tseng to prove log-concavity properties of the Tutte polynomial.