  Minimal $SU(5)$ Grand Unified models predict massless neutrinos and struggle
to achieve gauge coupling unification compatible with the observed lower limit
on the proton lifetime. Both of these issues can be resolved by embedding
minimal radiative neutrino mass models into $SU(5)$. We systematically analyze
the possible ways to realize radiative neutrino mass generation in $SU(5)$ and
provide a list of the minimal models. We find various models that have not been
considered in the literature and demonstrate the compatibility of radiative
neutrino masses with gauge coupling unification and proton decay for a new
class of models with vector-like fermions.
