  This article is a survey of recent work of the authors developing a new
approach to quantization based on the equivariance with respect to some Lie
group of symmetries. Examples are provided by conformal and projective
differential geometry: given a smooth manifold M endowed with a flat
conformal/projective structure, we establish a canonical isomorphism between
the space of symmetric contravariant tensor fields on M and the space of
differential operators on M. This leads to a notion of conformally/projectively
invariant star-product on $T^*M$.
