I will illustrate the most recent results on how to enhance triangulated categories (and exact functors) of geometric nature. We will then move to the problem of lifting equivalences between various triangulated categories and illustrate the new interplay between the theory of weakly approximable triangulated categories and the existing results about the uniqueness of enhancements. Applications to a generalization of a classical result by Rickard and to derived invariants of schemes will be discussed. The new results are joint works (partly in progress) with Alberto Canonaco, Amnon Neeman and Mattia Ornaghi.