Fourier-Mukai functors: derived vs dg categories

Fourier-Mukai functors play a distinct role in algebraic geometry.
Nevertheless a basic question is still open: are all exact functors
between the bounded derived categories of smooth projective varieties of
Fourier-Mukai type? We discuss the recent advances in the subject and study
the same question in the context of dg categories where the problem has been
settled by B. To\\en. In this talk we propose a simpler approach not