There has been a recent flurry of interest in the spectral theory of tensors and hypergraphs as new ideas have faithfully analogized spectral graph theory to hypergraphs. However, even in their simplest incarnation -- the homogeneous adjacency spectrum -- a large number of seemingly basic questions about hypergraph spectra remain out of reach. One of the problems that has yet to be resolved is the (asymptotically almost sure) spectrum of a random hypergraph in the Erdős-Rényi sense. We introduce the requisite theoretical framework and discuss some progress in this area that involves tools from commutative algebra, eigenvalue stability, and large deviations.