  We develop the natural tractor calculi associated to conformal and
  CR structures as a fundamental tool for the study of Fefferman's construction
of a canonical conformal class on the total space of a circle bundle over a
non--degenerate CR manifold of hypersurface type. In particular we construct
and treat the basic objects that relate the natural bundles and natural
operators on the two spaces. This is illustrated with several applications: We
prove that a number of conformally invariant overdetermined systems admit
non--trivial solutions on any Fefferman space. We show that the space of
conformal Killing fields on a Fefferman space naturally decomposes into a
direct sum of subspaces, which admit an interpretaion as solutions of certain
CR invariant PDE's. Finally we explicitly analyze the relation between tractor
calculus on a CR manifold and the complexified conformal tractor calculus on
its Fefferman space, thus obtaining a powerful computational tool for working
with the Fefferman construction.
