The Chow rings of matroids were constructed by Feichtner—Yuzvinsky as the Chow ring of smooth, but typically noncompact, toric varieties. They satisfy remarkable Hodge theoretic properties that are satisfied by a smooth projective variety. These special properties became the start of combinatorial Hodge theory, and played a key role in the resolution of the Heron-Rota-Welsh conjecture on log-concavity of characteristic polynomials of matroids, and the Dowling—Wilson top-heavy conjecture. I will survey some properties of Chow rings and K-rings of matroids, including the construction of an exceptional Hirzebruch—Riemann—Roch theorem. I will touch on some progress on open conjectures surrounding real-rootedness of Poincare polynomials of the Chow rings of matroids, and their connection to combinatorial commutative algebra.