  First we prove that the second adjoint of every bounded linear operator
taking values in a Dedekind $\sigma$-complete Banach lattice containing a copy
of $\ell_\infty$ is almost Dunford-Pettis. Next we extend some known results
concerning conditions on the Banach lattices $E$ and $F$ under which the
adjoint $T^\ast$ and the second adjoint $T^{\ast \ast}$ of any positive almost
Dunford-Pettis operator $T : E \longrightarrow F$ are almost Dunford-Pettis.
Finally, we prove when $T^\ast$ and $T^{\ast \ast}$ are almost Dunford-Pettis
for any (non necessarily almost Dunford-Pettis) $T$ that is either bounded,
regular, order bounded or weakly compact.
