Résumé : The packet routing problem plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In the $(\ell,k)$-routing problem, each node can send at most $\ell$ packets and receive at most $k$ packets. Permutation routing is the particular case $\ell=k=1$. In the $r$-central routing problem, all nodes at distance at most $r$ from a fixed node $v$ want to send a packet to $v$. In this article we study the permutation routing, the $r$-central routing and the general $(\ell,k)$-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the \emph{store-and-forward} $\Delta$-port model, and we consider both full and half-duplex networks. The main contributions are the following: \begin{itemize} \item[1.] Tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids. \item[2.] Tight $r$-central routing algorithms on triangular and hexagonal grids. \item[3.] Tight $(k,k)$-routing algorithms on square, triangular and hexagonal grids. \item[4.] Good approximation algorithms (in terms of running time) for $(\ell,k)$-routing on square, triangular and hexagonal grids, together with new lower bounds on the running time of any algorithm using shortest path routing. \end{itemize} \noindent All these algorithms are completely distributed, i.e. can be implemented independently at each node. Finally, we also formulate the $(\ell,k)$-routing problem as a \textsc{Weighted Edge Coloring} problem on bipartite graphs.