If f_* has a right adjoint, it must preserve colimits and hence be right-exact. Thus a necessary condition is that the higher derived functors vanish. In particular, when everything is affine and we have X=Spec B, Y=Spec A, then I believe the adjoint exists and is given by M \mapsto Hom_A(B,M).

If $f_\ast$ has a right adjoint, it must preserve colimits and hence be right-exact. Thus a necessary condition is that the higher derived functors vanish. In particular, when everything is affine and we have $X=\operatorname{Spec} B$ and $Y=\operatorname{Spec} A$, then I believe the adjoint exists and is given by $M \mapsto \operatorname{Hom}_A(B,M)$.