  It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
$c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx
27.23$ is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
  An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability.
