  Given a Fourier-Mukai functor $\Phi$ in the general setting of singular
schemes, under various hypotheses we provide both left and a right adjoints to
$\Phi$, and also give explicit formulas for them. These formulas are simple and
natural, and recover the usual formulas when the Fourier-Mukai kernel is a
perfect complex. This extends previous work of Anno and Logvinenko, and
Hernandez Ruiperez, Lopez Martin and Sancho de Salas, and has applications to
the twist autoequivalences of Donovan and Wemyss.
