In the framework of geometric quantization, it is common to define a quantization procedure as a linear bijection from the space of classical observables to a space of differential operators acting on wave functions (see [31]). In our setting, the space of observables (also called the space of Symbols) is the space of smooth functions on the cotangent bundle T ∗M of a manifold M , that are polynomial along the fibres. The space of differential operators D 1 2 (M) is made of differential operators acting on half-densities. It is known that there is no natural quantization procedure : the spaces of symbols and of differential operators are not isomorphic as representations of Diff(M). The idea of G-equivariant quantization is to reduce the set of (local) diffeomorphisms under consideration. If a Lie group G acts on M by local diffeomorphisms, this action can be lifted to symbols and to differential operators. A G-equivariant quantization was defined by P. Lecomte and V. Ovsienko in [21] as a G-module isomorphism from symbols to differential operators. They first considered the projective group PGL(m+1,R) acting locally on the manifold M = Rm by linear fractional transformations and defined the notion of projectively equivariant quantization. They proved the existence of such a quantization and its uniqueness, up to some natural normalization condition. In [11], the authors considered the group SO(p + 1, q + 1) acting on the space Rp+q or on a manifold endowed with a flat conformal structure. There again, the result was the existence and uniqueness of a conformally equivariant quantization. Over vector spaces, or manifolds endowed with flat structures, similar results were obtained for other type of differential operators (see [1]) or for other Lie groups. The first part of this presentation is a survey of these results. Unless otherwise stated, the results are based on a collaboration with F. Boniver (see [3] and [4])