Equivariant quantization, in the sense of C. Duval, P. Lecomte, and V. Ovsienko, developed as from 1996 in a quite small community. This procedure requires equivariance of the quantization map with respect to the action of a symmetry group G, or, on the infinitesimal level, with respect to the action of a Lie subalgebra of vector fields, is well-defined globally on manifolds endowed with a flat G-structure, and leads to invariant star-products. It has first been studied on vector spaces for the projective and conformal groups, and then extended in 2001 to arbitrary manifolds. In this setting, equivariance with respect to all diffeomorphisms has been restored by addition of a new argument. This has led to the concept of natural and projectively invariant quantization. Existence of such quantization maps has been investigated in several works that will be briefly explained.