Neutrino Masses and Mixing in Supersymmetric Models without R Parity

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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: θij m(ℓᵢ-)/m(ℓⱼ-) (i<j) and unusual order of magnitude predictions for neutrino mass ratios: m(νₑ)/m(νμ) ²θ₁₂; m(νᵢ)/m(ντ) 10⁻⁷ ²θᵢ₃ (i=1,2). Bounds from laboratory experiments exclude mντ 3MeV and the cosmological constraint that excludes mντ 100eV is not evaded. Neither the solar nor the atmospheric neutrino problems are likely to be solved by νμ-νₑ oscillations. These conclusions can be evaded if holomorphy plays an important role in the lepton Yukawa couplings.