  We prove the rationality and irreducibility of the moduli space of---what we
call---the endomorphism-general instanton vector bundles of arbitrary rank on
the projective space. In particular, we deduce the rationality of the moduli
spaces of rank-two mathematical instantons. This problem was first studied by
Hartshorne, Hirschowitz-Narasimhan in the late 1970s, and it has been
reiterated within the framework of the ICM 2018.
