Title: A survey on natural quantization

Equivariant quantization, in the sense of C.Duval, P.Lecomte, and V.Ovsienko, has developed as from 1996 in a rather small community. This quantization procedure requires equivariance of the quantization map with respect to the action of a symmetry group G, is well defined globally on manifolds endowed with a flat G-structure, and leads to invariant star-products.

It was first studied on vector spaces for the projective and the conformal groups, then extended in 2001 to arbitrary manifolds. In this setting, equivariance with respect to all diffeomorphisms has been restored, which has finally led to the concept of natural and projectively invariant quantization. Existence of such quantization maps has been investigated in several recent works, which will briefly be addressed in this talk.