  A plausible explanation for the lightness of neutrino masses is that
neutrinos are massless at tree level, with their mass (typically Majorana)
being generated radiatively at one or more loops. The new couplings, together
with the suppression coming from the loop factors, imply that the new degrees
of freedom cannot be too heavy (they are typically at the TeV scale).
Therefore, in these models there are no large mass hierarchies and they can be
tested using different searches, making their detailed phenomenological study
very appealing. In particular, the new particles can be searched for at
colliders and generically induce signals in lepton-flavor and lepton-number
violating processes (in the case of Majorana neutrinos), which are not
independent from reproducing correctly the neutrino masses and mixings. The
main focus of the review is on Majorana neutrinos. We order the allowed theory
space from three different perspectives: (i) using an effective operator
approach to lepton number violation, (ii) by the number of loops at which the
Weinberg operator is generated, (iii) within a given loop order, by the
possible irreducible topologies. We also discuss in more detail some popular
radiative models which involve qualitatively different features, revisiting
their most important phenomenological implications. Finally, we list some
promising avenues to pursue.
