This highly interesting paper develops a new method, called quasi-Lie schemes, which consist essentially of a system of nonautonomous differential equations equipped with a flow of diffeomorphisms transforming it into a Lie system. The main structural properties, like the existence of superposition principles, are analyzed. The method is seen to provide a geometrical insight absent from the formal approach, and from this new perspective further information is gained. As important examples that illustrate the potential of the method, the dissipative Milne-Pinney equations are treated, as well as a geometrical reformulation of some systems considered many years ago by A. M. Perelomov [Commun. Math. Phys. 63, 9–11 (1978; Zbl 0435.70013)]. The application to the quantum case enables the authors to give an explanation of the transformation properties of quantum analogues of physical systems.