This talk focuses on the structure of rational inner functions (i.e. several-variable generalizations of finite Blaschke products) with singularities on the $d$-torus. In the two-variable setting, one can quantitatively measure ``how singular'' such functions are using information pulled from their zero sets and unimodular level sets. In the $d$-variable setting, rational inner function behavior is much more complicated. While some results about zero sets and integrability generalize, a host of examples illustrate obstructions to a general theory. This is joint work with Alan Sola and James Pascoe.