After outlining some popular ways to go beyond the standard model so as to generate non-zero but tiny neutrino masses, I focus on several typical seesaw mechanisms and discuss how to get a balance between their theoretical naturalness and their experimental testability. Neutrinos are assumed or required to be massless in the SM, just because the structure of the SM itself is too simple to accommodate massive neutrinos.• Two fundamentals of the SM are the SU (2) L × U (1) Y gauge symmetry and the Lorentz invariance. Both of them are mandatory to guarantee that the SM is a consistent quantum field theory.• The particle content of the SM is rather economical. There are no right-handed neutrinos in the SM, so a Dirac neutrino mass term is not allowed. There is only one Higgs doublet, so a gauge-invariant Majorana mass term is forbidden.• The SM is a renormalizable quantum field theory. Hence an effective dimension-5 operator, which may give each neutrino a Majorana mass, is absent. In other words, the SM accidently possesses the (B − L) symmetry which assures three known neutrinos to be exactly massless.But today's experiments have convincingly indicated the existence of neutrino oscillations. 1) This quantum phenomenon can appear if and only if neutrinos are massive and lepton flavors are mixed, and thus it is a kind of new physics beyond the SM. To generate non-zero but tiny neutrino masses, one or more of the abovementioned constraints on the SM must be abandoned or relaxed. It is certainly intolerable to abandon the gauge symmetry and Lorentz invariance; 2) otherwise, one would be led astray. Given the framework of the SM as a consistent field theory, its particle content can be modified and (or) its renormalizability can be abandoned to accommodate massive neutrinos. There are several ways to this goal. * )