  A covariant Poisson bracket and an associated covariant star product in the
sense of deformation quantization are defined on the algebra of tensor-valued
differential forms on a symplectic manifold, as a generalization of similar
structures that were recently defined on the algebra of (scalar-valued)
differential forms. A covariant star product of arbitrary smooth tensor fields
is obtained as a special case. Finally, we study covariant star products on a
more general Poisson manifold with a linear connection, first for smooth
functions and then for smooth tensor fields of any type. Some observations on
possible applications of the covariant star products to gravity and gauge
theory are made.
