  We provide a complete systematic classification of all two-loop realizations
of the dimension four operator for Dirac neutrino masses. Our classification is
multi-layered, starting first with a classification in terms of all possible
distinct two-loop topologies. Then we discuss the possible diagrams for each
topology. Model-diagrams originating from each diagram are then considered. The
criterion for genuineness is also defined and discussed at length. Finally, as
examples, we construct two explicit models which also serve to highlight the
intimate connection between the Dirac nature of neutrinos and dark matter
stability.
