  We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times
S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology
and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli
subgroup, determine the abelianisations, and relate our results to the group of
homotopy equivalences of these manifolds.
