  We study instanton bundles $E$ on $\mathbb{P}^1\times \mathbb{P}^1 \times
\mathbb{P}^1$. We construct two different monads which are the analog of the
monads for instanton bundles on $\mathbb P^3$ and on the flag threefold
$F(0,1,2)$. We characterize the Gieseker semistable cases and we prove the
existence of $\mu$-stable instanton bundles generically trivial on the lines
for any possible $c_2(E)$. We also study the locus of jumping lines.
