  A quantization over a manifold can be seen as a way to construct a
differential operator with prescribed principal symbol. The quantization map is
moreover required to be a linear bijection. It is known that there is in
general no natural quantization procedure. However, considering manifolds
endowed with additional structures, such as projective or pseudo-conformal
structures, one can seek for quantizations that depend on this additional
structure and that are natural if the dependence with respect to the structure
is taken into account.
  The question of existence of such a quantization was addressed in a series of
papers in the context of projective geometry, using the framework of
Thomas-Whitehead connections (see Bordemann, Hansoul and Fox). Recently, we
recovered these existence results, using the theory of Cartan projective
connections.
  In the present work, we show that our method can be adapted to
pseudo-conformal geometry to yield the so-called natural and conformally
invariant quantization for arbitrary symbols, still outside some critical
situations. Our method is general enough to analyze the problem of invariant
quantizations in the context of manifolds with irreducible parabolic
geometries.
