Summary: We give two, new upper bounds for oblivious permutation routing on the mesh networks: Let N be the total number of processors in each $\surd \surd $mesh. One is an $O(N0.75)$ algorithm on the two-dimensional, N $\times N$ mesh with constant queue-size. This is the first algorithm which improves substantially the trivial $O(N)$ bound for oblivious routing in the mesh $\surd \surd $networks with constant queue-size. The other is a $1.16 N + o( N )$ algorithm on the three-dimensional, N1/$3 \times N1/3 \times N1$/3 mesh with unlimited queue-size. This algorithm allows at most three bends in the path of each packet. If the number of bends is restricted to minimal, i.e., at most two, then the bound jumps to $\Omega $(N2/3) as was shown in ESA'97. Communicated by T. Nishizeki, R. Tamassia and D. Wagner: