  This paper surveys the recent advances concerning the relations between
triangulated (or derived) categories and their dg enhancements. We explain when
some interesting triangulated categories arising in algebraic geometry have a
unique dg enhancement. This is the case, for example, for the unbounded derived
category of quasi-coherent sheaves on an algebraic stack or for its full
triangulated subcategory of perfect complexes. Moreover we give an account of
the recent results about the possibility to lift exact functors between the
bounded derived categories of coherent sheaves on smooth schemes to dg
(quasi-)functors.
