  We present a Donaldson-Witten type field theory in eight dimensions on
manifolds with $Spin(7)$ holonomy. We prove that the stress tensor is BRST
exact for metric variations preserving the holonomy and we give the invariants
for this class of variations. In six and seven dimensions we propose similar
theories on Calabi-Yau threefolds and manifolds of $G_2$ holonomy respectively.
We point out that these theories arise by considering supersymmetric Yang-Mills
theory defined on such manifolds. The theories are invariant under metric
variations preserving the holonomy structure without the need for twisting.
This statement is a higher dimensional analogue of the fact that
Donaldson-Witten field theory on hyper-K\"ahler 4-manifolds is topological
without twisting. Higher dimensional analogues of Floer cohomology are briefly
outlined. All of these theories arise naturally within the context of string
theory.
