  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
bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of
paths and (even) cycles, was essentially determined as well. Improving the
results of DeBiasio, Gy\'arf\'as, Krueger, Ruszink\'o, and S\'ark\"ozy, in this
paper we determine asymptotically the $4$-colour bipartite Ramsey number of
paths and cycles. We also provide new upper bounds on the $k$-colour bipartite
Ramsey numbers of paths and cycles which are close to being tight.
