  We propose a notion of instanton bundle (called $H$-instanton bundle) on any
projective variety of dimension three polarized by a very ample divisor $H$,
that naturally generalizes the ones on $\mathbb{P}^3$ and on the flag threefold
$F(0,1,2)$. We discuss the cases of Veronese and Fano threefolds. Then we deal
with $H$-instanton bundles $\mathcal{E}$ on three-dimensional rational normal
scrolls $S(a_0,a_1,a_2)$. We give a monadic description of $H$-instanton
bundles and we prove the existence of $\mu$-stable $H$-instanton bundles on
$S(a_0,a_1,a_2)$ for any admissible charge $k=c_2(\mathcal{E})H$. Then we deal
in more detail with $S(a,a,b)$ and $S(a_0,a_1,a_2)$ with $a_0+a_1>a_2$ and even
degree. Finally we describe a nice component of the moduli space of
$\mu$-stable bundles whose points represent $H$-instantons.
