  We study systematically the decomposition of the Weinberg operator at
three-loop order. There are more than four thousand connected topologies.
However, the vast majority of these are infinite corrections to lower order
neutrino mass diagrams and only a very small percentage yields models for which
the three-loop diagrams are the leading order contribution to the neutrino mass
matrix. We identify 73 topologies that can lead to genuine three-loop models
with fermions and scalars, i.e. models for which lower order diagrams are
automatically absent without the need to invoke additional symmetries. The 73
genuine topologies can be divided into two sub-classes: Normal genuine ones (44
cases) and special genuine topologies (29 cases). The latter are a special
class of topologies, which can lead to genuine diagrams only for very specific
choices of fields. The genuine topologies generate 374 diagrams in the weak
basis, which can be reduced to only 30 distinct diagrams in the mass eigenstate
basis. We also discuss how all the mass eigenstate diagrams can be described in
terms of only five master integrals. We present some concrete models and for
two of them we give numerical estimates for the typical size of neutrino masses
they generate. Our results can be readily applied to construct other $d=5$
neutrino mass models with three loops.
