Fourier--Mukai functors in geometric contexts

Fourier-Mukai functors play a distinct role in algebraic geometry. Nevertheless two questions remained open: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? Is the Fourier-Mukai kernel unique? We will provide a negative answer to the latter problem while we will give results related to the first one and extending previous results by Lunts, Orlov and Ballard. Along the way, we will show that, in geometric contexts, full functors are faithful as well. This is a joint work in collaboration with A. Canonaco and D. Orlov.