  The concept of conformally equivariant quantizations was introduced by Duval,
Lecomte and Ovsienko in \cite{DLO} for manifolds endowed with flat conformal
structures. They obtained results of existence and uniqueness (up to
normalization) of such a quantization procedure. A natural generalization of
this concept is to seek for a quantization procedure, over a manifold $M$, that
depends on a pseudo-Riemannian metric, is natural and is invariant with respect
to a conformal change of the metric. The existence of such a procedure was
conjectured by P. Lecomte in \cite{Leconj} and proved by C. Duval and V.
Ovsienko in \cite{DO1} for symbols of degree at most 2 and by S. Loubon Djounga
in \cite{Loubon} for symbols of degree 3. In two recent papers \cite{MR,MR1},
we investigated the question of existence of projectively equivariant
quantizations using the framework of Cartan connections. Here we will show how
the formalism developed in these works adapts in order to deal with the
conformally equivariant quantization for symbols of degree at most 3. This will
allow us to easily recover the results of \cite{DO1} and \cite{Loubon}. We will
then show how it can be modified in order to prove the existence of conformally
equivariant quantizations for symbols of degree 4.
