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. We first survey the existing results in the literature about packet routing, with special emphasis on $(ell,k)$-routing on plane grids. Our main contributions are the following: 1. Tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids. 2. Tight $r$-central routing algorithms on triangular and hexagonal grids. 3. Tight $(k,k)$-routing algorithms on square, triangular and hexagonal grids. 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. These algorithms are all 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.
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