I discuss dynamical generation of neutrino masses in unconventional scenarios where the background space-time geometry plays a crucial role. I discuss two types of backgrounds: (i) Lorentz Violating and (ii) Geometries with Torsion. In the former case, the violation of Lorentz symmetry, at a scale M, may be viewed as a catalyst for mass generation and induced flavour oscillations among neutrino species, which survive the limit of M taken to infinity, leading to a hierarchy among neutrino masses. In the latter case, the (totally antisymmetric components of the) torsion degrees of freedom correspond to a pseudoscalar axion field in four space-time dimensions. This field is assumed to be mixed, through non-diagonal kinetic terms, with ordinary axion fields that may exist in the theory for other reasons and couple to neutrinos with chirality changing Yukawa couplings. The torsion-ordinary-axion-field mixing is responsible, through higher-loop anomalous graphs, for the dynamical generation of Majorana masses. The latter scenarios can also be realised in some (compactified) string theory models, where the (totally antisymmetric) torsion is played by the field strength of the spin-one antisymmetric tensor (Kalb-Ramond) field, which exists in the gravitational string multiplet.