  Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this
note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$
of $I$. We compute its regularity and the universal Gr\"obner basis of its
defining equations; interestingly, both of them are described in terms of the
combinatorics of $G$. We apply these ideas to study the regularity of the
powers of $I$. For any $s \ge \text{match}(G)+\lvert E(G) \rvert +1$ we prove
that $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.
