  We develop a spinorial description of CR structures of arbitrary codimension.
More precisely, we characterize almost CR structures of arbitrary codimension
on (Riemannian) manifolds by the existence of a Spin$^{c, r}$ structure
carrying a partially pure spinor field. We study various integrability
conditions of the almost CR structure in our spinorial setup, including the
classical integrability of a CR structure as well as those implied by
Killing-type conditions on the partially pure spinor field. In the codimension
one case, we develop a spinorial description of strictly pseudoconvex CR
manifolds, metric contact manifolds and Sasakian manifolds. Finally, we study
hypersurfaces of Kaehler manifolds via partially pure Spin$^c$ spinors.
