This talk will discuss some recent work on the spectra of discrete Laplace operators coming from boundary maps in a simplicial complex. For two families of simplicial complexes, the chessboard complexes (studied by J. Friedman and P. Hanlon) and matroid complexes (studied by W. Kook, D. Stanton and myself), these spectra are known to be integral, and interpretations of the spectra have been given. Why these particular complexes should have integral spectra is still mysterious, as is the connection to spectra of Laplacians on Riemannian manifolds. One corollary to the interpretation of the spectra for matroid complexes may be paraphrased as stating that one can "hear" the chromatic polynomial of a graph.