  We give a proof of Kontsevich's formality theorem for a general manifold
using Fedosov resolutions of algebras of polydifferential operators and
polyvector fields. The main advantage of our construction of the formality
quasi-isomorphism is that it is based on the use of covariant tensors unlike
Kontsevich's original proof, which is based on $\infty$-jets of
polydifferential operators and polyvector fields. Using our construction we
prove that if a group G acts smoothly on a manifold M and M admits a
G-invariant affine connection then there exists a G-equivariant
quasi-isomorphism of formality. This result implies that if a manifold M is
equipped with a smooth action of a finite or compact group G or equipped with a
free action of a Lie group G then M admits a G-equivariant formality
quasi-isomorphism. In particular, this gives a solution of the deformation
quantization problem for an arbitrary Poisson orbifold.
