We generalize the Poisson–Lie T-duality by making use of the structure of the affine Poisson group which is the concept introduced some time ago in Poisson geometry as a generalization of the Poisson–Lie group. We also introduce a new notion of an affine quasi-Poisson group and show that it gives rise to a still more general T-duality framework. We establish for a class of examples that this new T-duality is compatible with the renormalization group flow.