Neutrino properties and SUSY without $R$-parity

In supersymmetric models without $R$-parity neutrinos naturally become massive and mix with each other. We explore the predictions of a very restricted model with only three free parameters and find that this model naturally yields masses and mixing angles compatible with experimental results from solar and atmospheric neutrino experiments. Furthermore, there is a tiny region in parameter space where the solution to the solar neutrino puzzle is compatible with either the LSND result or the existence of significant hot dark matter neutrinos.