On the cohomology of Torelli groups. II

We describe the ring structure of the rational cohomology of the Torelli groups of the manifolds ${\mathrm{\#}}^g{S}^n\times S^n$ in a stable range, for $2n\ge 6$. Some of our results are also valid for $2n=2$, where they are closely related to unpublished results of Kawazumi and Morita.