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A celebrated result by Orlov states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of geometric origin, i.e. it is a Fourier–Mukai functor. In this paper we prove that any smooth projective variety of dimension 
$\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier–Mukai functor between smooth projective schemes.
We consider the problem of preservation of stability under the Fourier–Mukai transform ℱℰ:D(X)→D(Y ) on an abelian surface and a K3 surface. If Y is the moduli space of μ-stable sheaves on X with respect to a polarization H, we have a canonical polarization 
on Y and we have a correspondence between (X,H) and 
. We show that the stability with respect to these polarizations is preserved under ℱℰ, if the degree of stable sheaves on X is sufficiently large.
