One way to define a matroid is via its base polytope.  From this point of view, some matroid invariants easily have geometric interpretations: e.g., the number of bases is the number of vertices of the polytope.  It turns out that most interesting matroid invariants are also geometric, in the sense that they are valuative.  Valuative invariants are linear functionals on a suitable graded abelian group of matroids, modulo valuative equivalence.  Remarkably, this group can be identified with the Chow ring of the stellahedral or permutohedraltoric variety, depending on whether or not one considers matroids with loops, which goes a long way to explaining what valuative invariants look like. Informally, a valuative invariant is one that respects inclusion-exclusion relations on (matroid) polytopes. This talk is based on work of Eur-Huh-Larson (2023), Hampe (2017), Ferroni-Schröter (2022), Derksen-Fink (2009), Speyer (2008) and others.