  Given associative unital algebras $A$ and $B$ and a complex $T^\bullet$ of
$B-A-$bi\-modules, we give necessary and sufficient conditions for the total
derived functors, $\Rh_A(T^\bullet,?):\D(A)\longrightarrow\D(B)$ and
$?\Lt_BT^\bullet:\D(B)\longrightarrow\D(A)$, to be fully faithful. We also give
criteria for these functors to be one of the fully faithful functors appearing
in a recollement of derived categories. In the case when $T^\bullet$ is just a
$B-A-$bimodule, we connect the results with (infinite dimensional) tilting
theory and show that some open question on the fully faithfulness of
$\Rh_A(T,?)$ is related to the classical Wakamatsu tilting problem.
