This talk will be about a pair of related matrix integrals, the Harish-Chandra/Itzykson-Zuber integral and the Brezin-Gross-Witten integral, which play an important role in random matrix theory, representation theory, and mathematical physics. While these integrals cannot be exactly evaluated, an old conjecture says that they admit asymptotic expansions whose coefficients are themselves generating functions for some unspecified combinatorial invariants of compact Riemann surfaces (or smooth projective curves).