  We show that, in some cases, the projective and the injective tensor products
of two Banach spaces do not have the Dunford-Pettis property (DPP). As a
consequence, we obtain that $(c_0\hat{\otimes}_\pi c_0)^{**}$ fails the DPP.
Since $(c_0\hat{\otimes}_\pi c_0)^{*}$ does enjoy it, this provides a new space
with the DPP whose dual fails to have it. We also prove that, if $E$ and $F$
are ${\mathscr L}_1$-spaces, then $E\hat{\otimes}_\epsilon F$ has the DPP if
and only if both $E$ and $F$ have the Schur property. Other results and
examples are given.
