In this learning seminar, we have been converting questions about face counts of polytopes into questions about the cohomology of toric varieties. In the case of toric varieties associated to a (nice enough) fan, the cohomology is isomorphic to a combinatorial Chow ring. As their name suggests, the generators and relations of these rings can be described directly from the combinatorics of the fan. We will describe these rings, and give several detailed examples.