  This paper is devoted to the systematic investigation of the cone
construction for Riemannian $G$ manifolds M, endowed with an invariant metric
connection with skew torsion $\nabla^c$, a `characteristic connection'. We show
how to define a $\bar G$ structure on the cone $\bar M=M\x \R^+$ with a cone
metric, and we prove that a Killing spinor with torsion on $M$ induces a spinor
on $\bar M$ that is parallel w.\,r.\,t. the characteristic connection of the
$\bar G$ structure. We establish the explicit correspondence between classes of
metric almost contact structures on $M$ and almost hermitian classes on $\bar
M$, resp. between classes of $G_2$ structures on $M$ and $\Spin(7)$ structures
on $\bar M$. Examples illustrate how this `cone correspondence with torsion'
works in practice.
