  We introduce a covariant non-commutative deformation of 3+1-dimensional
conformal field theory. The deformation depends on a short-distance scale
\ell_p, and thus breaks scale invariance, but preserves all space-time
isometries. The non-commutative algebra is defined on space-times with non-zero
constant curvature, i.e. dS_4 or AdS_4. The construction makes essential use of
the representation of CFT tensor operators as polynomials in an auxiliary
polarization tensor. The polarization tensor takes active part in the
non-commutative algebra, which for dS_4 takes the form of so(5,1), while for
AdS_4 it assembles into so(4,2). The structure of the non-commutative
correlation functions hints that the deformed theory contains gravitational
interactions and a Regge-like trajectory of higher spin excitations.
