Spinorial description of SU(3)- and G2-manifolds

We present a uniform description of SU(3)-structures in dimension 6 as well as G2-structures in dimension 7 in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor.