We investigate the compactification of D=11 supergravity to D=5,4,3, on compact manifolds of holonomy \(SU(3)\) (Calabi-Yau), \(G_2\), and \(Spin(7)\), respectively, making use of examples of the latter two cases found recently by Joyce. In each case the lower dimensional theory is a Maxwell/Einstein supergravity theory. We find evidence for an equivalence, in certain cases, with heterotic string compactifications from D=10 to D=5,4,3, on compact manifolds of holonomy \(SU(2)\) (\(K_3\times S^1\)), \(SU(3)\), and \(G_2\), respectively. Calabi-Yau manifolds with Hodge numbers \(h_{1,1}=h_{1,2}=19\) play a significant role in the proposed equivalences.