  If neutrino masses have a radiative origin, their smallness can be naturally
understood even when lepton number violation occurs near the weak scale. We
analyze a specific model of this type wherein the neutrino masses arise as
two--loop radiative corrections. We show that the model admits the near
bimaximal mixing pattern suggested by the current neutrino oscillation data.
Unlike the conventional seesaw models, these two--loop models can be directly
tested in lepton flavor violating decays tau -> 3 mu and mu -> e + gamma as
well as at colliders by the direct observation of charged scalars needed for
the mass generation. It is shown that consistency with the neutrino oscillation
data requires that the leptonic rare decays should be within reach of
forthcoming experiments and that the charged scalars are likely to be within
reach of the LHC.
