  We systematically analyze the $d=5$ Weinberg operator at 2-loop order. Using
a diagrammatic approach, we identify two different interesting categories of
neutrino mass models: (i) Genuine 2-loop models for which both, tree-level and
1-loop contributions, are guaranteed to be absent. And (ii) finite 2-loop
diagrams, which correspond to the 1-loop generation of some particular vertex
appearing in a given 1-loop neutrino mass model, thus being effectively 2-loop.
From the large list of all possible 2-loop diagrams, the vast majority are
infinite corrections to lower order neutrino mass models and only a moderately
small number of diagrams fall into these two interesting classes. Moreover, all
diagrams in class (i) are just variations of three basic diagrams, with
examples discussed in the literature before. Similarly, we also show that class
(ii) diagrams consists of only variations of these three plus two more basic
diagrams. Finally, we show how our results can be consistently and readily used
in order to construct two-loop neutrino mass models.
