  The genus of the binomial random graph $G(n,p)$ is well understood for a wide
range of $p=p(n)$. Recently, the study of the genus of the random bipartite
graph $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was
initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are
comparable in size and $p=p(n_1,n_2)$ is significantly larger than
$(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar
behaviour to that of the binomial random graph.
  In this paper we show that there is a threshold for planarity of the random
bipartite graph at $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close
to this threshold, extending the results of Mohar and Ying. It turns out that
there is qualitatively different behaviour in the case where $n_1$ and $n_2$
are comparable, when whp the genus is linear in the number of edges, than in
the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus
behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately
chosen $q=q(p,n_1,n_2)$.
