The aim of this talk consists in obtaining a more general notion of deformation through symmetry, by using formality theory. First, we show an alternative global version of the twisting procedure used by Dolgushev to prove formality theorems. This allows us to describe the maps of Fedosov resolutions in terms of a twist of the fiberwise quasi-isomorphisms. In the second part, using the above global version of the twisting procedure, we provide a quantization via formality of Poisson actions of a triangular Lie algebra on a smooth manifold. The talk is based on joint works with N. de Kleijn.