  We discuss higher-dimensional gravitational instantons by studying
appropriate self-duality equations for the spin connection. In seven and in
eight dimensions, the corresponding spaces admit a covariantly constant spinor
and have holonomies in G_2 and Spin(7), respectively. We find a non-compact
solution to the self-duality equations in eight dimensions in which the
self-dual space has an elliptically-fibered structure.
