The Harish-Chandra-Itzykson-Zuber integral is a remarkable special function which plays a key role in random matrix theory, where it enters into the description of the spectra of coupled random matrices, Hermitian Wigner matrices, and complex sample covariance matrices. As shown by Guionnet and Zeitouni, the leading asymptotics of the HCIZ integral can be characterized as the solution to a certain variational problem. I will present joint work with I. Goulden and M. Guay-Paquet which relates the HCIZ integral to a classical topic in enumerative geometry, namely counting branched covers of the Riemann sphere with specified branch points and monodromies. In particular, the HCIZ integral can be viewed as a generating function for a desymmetrized version of the double Hurwitz numbers introduced by Okounkov.