  The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by
taking two partition classes of size $n$ and including each edge between them
independently with probability $p$. It is known that this model exhibits a
similar phase transition as that of the binomial random graph $G(n,p)$ as $p$
passes the critical point of $\frac{1}{n}$. We study the component structure of
this model near to the critical point. We show that, as with $G(n,p)$, for an
appropriate range of $p$ there is a unique `giant' component and we determine
asymptotically its order and excess. We also give more precise results for the
distribution of the number of components of a fixed order in this range of $p$.
These results rely on new bounds for the number of bipartite graphs with a
fixed number of vertices and edges, which we also derive.
