  In this paper we show that the Bishop-Phelps-Bollob\'as theorem holds for
$\mathcal{L}(L_1(\mu), L_1(\nu))$ for all measures $\mu$ and $\nu$ and also
holds for $\mathcal{L}(L_1(\mu),L_\infty(\nu))$ for every arbitrary measure
$\mu$ and every localizable measure $\nu$. Finally, we show that the
Bishop-Phelps-Bollob\'as theorem holds for two classes of bounded linear
operators from a real $L_1(\mu)$ into a real $C(K)$ if $\mu$ is a finite
measure and $K$ is a compact Hausdorff space. In particular, one of the classes
includes all Bochner representable operators and all weakly compact operators.
