Neutrino masses and mixing in supersymmetric models without R parity

We study neutrino masses and mixing in supersymmetric models without R parity and with generic soft Supersymmetry breaking terms. Neutrinos acquire mass from various sources: tree level neutrino-neutralino mixing, loop effects and non-renormalizable operators. Abelian horizontal symmetries (invoked to explain the smallness and hierarchy in quark parameters) replace R parity in suppressing neutrino masses. We find lower bounds on the mixing angles: sin theta(ij) greater than or similar to m(l(i)(-))/m(l(j)(-)) (i < j) and unusual order of magnitude predictions for neutrino mass ratios: m(nu(e))/m(nu(mu)) similar to sin(2) theta(12); m(nu(i))/m(nu(tau)) similar to 10(-7) sin(2) theta(i3) (i = 1, 2). Bounds from laboratory experiments exclude m(nu tau) greater than or similar to 3 MeV and cosmological constraints exclude m(nu tau) greater than or similar to 100 eV. Neither the solar nor the atmospheric neutrino problems are likely to be solved by nu(mu)-nu(e) oscillations. These conclusions can be evaded if holomorphy plays an important role in the lepton Yukawa couplings.