  We examine the asymptotics of the moments of characteristic polynomials of
$N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix
Theory, in the limit as $N\to\infty$. We focus in particular on the Gaussian
Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a
novel approach to calculate asymptotic formulae for the moments, enabling us to
uncover subtle structure not apparent in previous approaches.
