Abstract : 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 $r$-centralrouting problem, all nodes at distance at most $r$ from a fixed node $v$ want to send a packet to $v$.Here we survey the results on 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 assume the \\emphstore-and-forward Δ-port model, and we consider both full and half-duplex networks.