  Let $C_{n,g}$ be the number of rooted cubic maps with $2n$ vertices on the
orientable surface of genus $g$. We show that the sequence $(C_{n,g}:g\ge 0)$
is asymptotically normal with mean and variance asymptotic to $(1/2)(n-\ln n)$
and $(1/4)\ln n$, respectively. We derive an asymptotic expression for
$C_{n,g}$ when $(n-2g)/\ln n$ lies in any closed subinterval of $(0,2)$. Using
rotation systems and Bender's theorem about generating functions with
fast-growing coefficients, we derive simple asymptotic expressions for the
numbers of rooted regular maps, disregarding the genus. In particular, we show
that the number of rooted cubic maps with $2n$ vertices, disregarding the
genus, is asymptotic to $\frac{3}{\pi}\,n!6^n$.
