The notion of a vertex algebra is closely related to that of a commutative algebra. The latter provide the simplest examples of the former and, conversely, a general vertex algebra can be understood as a singular commutative algebra whose multiplication map depends on a parameter. This relation admits a natural quantization where the notion of a vertex algebra is replaced by that of quantum vertex algebra, which was introduced by Etingof and Kazhdan over 25 years ago. In this quantized story, commutative algebras grow up to become almost commutative algebras in the sense of quantum groups — their product and opposite product are intertwined by a solution of the quantum Yang–Baxter equation. The goal of this talk is to expand on this connection by explaining how to construct quantum vertex algebras within the category of representations of a distinguished class of quantum groups. This construction is a generalization of Majid's transmutation theory for quasitriangular Hopf algebras, and comes with a functor from the category of representations of the underlying quantum group to the category of modules over the resulting quantum vertex algebra. When applied to the Yangian of a simple Lie algebra, this yields a uniform construction of the Etingof–Kazhdan quantum affine vertex algebra whose structure maps are given by closed formulas. This is based on joint work with Alex Weekes and Matt Rupert.