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 (ℓ,k)-routing problem, each node can send at most ℓ packets and receive at most k packets. Permutation routing is the particular case ℓ=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 (ℓ,k)-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the \emph{store-and-forward} Δ-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 (ℓ,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 (ℓ,k)-routing problem as a \textsc{Weighted Edge Coloring} problem on bipartite graphs