We show improved bounds for 1-1 routing and sorting on multi-dimensional meshes and tori. In particular, we give a fairly simple deterministic algorithm for sorting on the d-dimensional mesh of side length n that achieves a running time of 3dn/2+o(n) without making any copies of the elements. We give deterministic algorithms with running times of 5dn/4+ o(n) and 3dn/4+ o(n) for the d-dimensional mesh and torus, respectively, that make one copy of each element. We also show lower bounds for sorting with respect to a large class of indexing schemes, under a model of the mesh where each processor can hold an arbitrary number of packets. Finally, we describe algorithms for permutation routing whose running times come very close to the diameter lower bound.