  We propose a general definition of mathematical instanton bundle with given
charge on any Fano threefold extending the classical definitions on $\mathbb
P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case
of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of
instanton bundles satisfying some important extra properties: moreover, we also
show that they correspond to smooth points of a component of the moduli space.
