We consider the partition function from a random Hermitian matrix model, $$ Z_{N} = \int e^{ - N { \rm{ Tr }} \left( M^2/2 + \sum_{\ell = 1}^{2 \nu} t_{\ell} M^{\ell} \right)} \ dM $$ where the parameters $t_{\ell}$, $\ell = 1, \ldots, 2 \nu$ are real, with $t_{2\nu} > 0$, the integral is taken over the space of $N \times N$ Hermitian matrices, and $d M$ is Lebesgue measure on the matrix entries. We prove that for ${\bf t}= (t_{1}, \ldots, t_{2 \nu})$ in a small but fixed sized neighborhood of
${\bf t} = {\bf 0}$, $\log{ Z_{N}}$ possesses an asymptotic expansion in even powers of $N$, $$ \log{ Z_{N}} = N^{2} e_{0}({\bf t}) + e_{1}({\bf t}) + N^{-2} e_{2}({\bf t}) + \cdots.$$   In the above expansion, each coefficient
$e_{j}({\bf t})$ is an analytic function of ${\bf t}$, in some neighborhood of ${\bf t} = {\bf 0}$. Amongst other things, this
represents a proof of the strong Szeg\H{o} limit theorem for Hankel determinants, for a wide class of given measures on the real line. In the talk we will explain how Riemann--Hilbert techniques are used to establish this result.