  We give a spinorial set of Hamiltonian variables for General Relativity in
any dimension greater than 2. This approach involves a study of the algebraic
properties of spinors in higher dimension, and of the elimination of
second-class constraints from the Hamiltonian theory. In four dimensions, when
restricted to the positive spin-bundle, these variables reduce to the standard
Ashtekar variables. In higher dimensions, the theory can either be reduced to a
spinorial version of the ADM formalism, or can be left in a more general form
which seems useful for the investigation of some spinorial problems such as
Riemannian manifolds with reduced holonomy group. In dimensions $0 \pmod 4$,
the theory may be recast solely in terms of structures on the positive
spin-bundle $\mathbb{V}^+$, but such a reduction does not seem possible in
dimensions $2 \pmod 4$, due to algebraic properties of spinors in these
dimensions.
