  We present simple and predictive scenarios for neutrino masses which lead to
a bimaximal pattern of mixings that apparently provides the best fit to the
current solar and atmospheric neutrino data. It is shown that an approximate
global L_e-L_\mu-L_\tau symmetry, broken only by Planck scale effects, can
naturally explain the bimaximal mixing pattern. In one scenario, the solar
neutrino oscillation parameters are induced entirely through radiative
renormalization group effects. A one parameter model describing all neutrino
data is presented, which predicts U_{e3} = \sqrt{m_e/m_\tau} \simeq 0.017. A
second two parameter scheme predicts U_{e3} to be vanishingly small. We also
show how these scenarios can be extended to accommodate the LSND observations
within a 3+1 neutrino oscillation scheme by including a mirror sector of
particles and forces.
