  We classify globally generated vector bundles on $\mathbf{P}^1 \times
\mathbf{P}^1 \times \mathbf{P}^1$ with small first Chern class, i.e. $c_1=
(a_1, a_2, a_3)$, $a_i \le 2$. Our main method is to investigate the associated
smooth curves to globally generated vector bundles via the Hartshorne-Serre
correspondence.
