  This paper presents an overview of the derivation and significance of
recently derived conservation laws for the matrix moments of Hermitean random
matrices with dominant exponential weights that may be either even or odd. This
is based on a detailed asymptotic analysis of the partition function for these
unitary ensembles and their scaling limits. As a particular application we
derive closed form expressions for the coefficients of the genus expansion for
the associated free energy in a particular class of dominant even weights.
These coefficients are generating functions for enumerating g-maps, related to
graphical combinatorics on Riemann surfaces. This generalizes and resolves a
30+ year old conjecture in the physics literature related to quantum gravity.
