  The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least
integer $n$ for which every $k$-edge-coloured complete bipartite graph
$K_{n,n}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey
numbers was initiated, over 40 years ago, by Faudree and Schelp and,
independently, by Gy\'arf\'as and Lehel, who determined the $2$-colour Ramsey
number of paths. In this paper we determine asymptotically the $3$-colour
bipartite Ramsey number of paths and (even) cycles.
