  Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all
graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it
is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph
$R_n$ be sampled uniformly from the graphs in $\mathcal{E}^g$ on vertex set
$[n]=\{1,\ldots,n\}$. Observe that if $g(n)$ is 0 then $R_n$ is a random planar
graph, and if $g(n)$ is sufficiently large then $R_n$ is a binomial random
graph $G(n,\tfrac12)$. We investigate typical properties of $R_n$. We find that
for \emph{every} genus function $g$, with high probability at most one
component of $R_n$ is non-planar. In contrast, we find a transition for example
for connectivity: if $g$ is non-decreasing and $g(n) = O(n/\log n)$ then
$\liminf_{n \to \infty} \mathbb{P}(R_n \mbox{ is connected}) < 1$, and if $g(n)
\gg n$ then with high probability $R_n$ is connected. These results also hold
when we consider orientable and non-orientable surfaces separately. We also
investigate random graphs sampled uniformly from the `hereditary part' or the
`minor-closed' part of $\mathcal{E}^g$, and briefly consider corresponding
results for unlabelled graphs.
