I will report on a joint work with Mattia Ornaghi and Paolo Stellari, where we prove that, over an arbitrary commutative ring, the localizations with respect to quasi-equivalences of the categories of dg (differential graded) categories and of A∞ categories are equivalent. The equivalence holds not only as ordinary categories, but even as ∞-categories; moreover, one can replace (strictly unital) A∞ categories with unital or cohomologically unital ones. We also show that the internal Homs in the homotopy category of dg categories can be realized as suitable dg categories of A∞ functors.