Theoretical ideas for generating neutrino mass often also predict new interactions for the neutrino.  Such non-standard interactions (NSI) will modify neutrino oscillation probabilities, especially during propagation through matter.  This talk will outline various experimental constraints and future tests of such NSI.  Light charged scalars present in the Zee model of neutrino mass generation will be shown to lead to "Zee bursts", analogous to Glashow resonance, which may be probed at IceCube-Gen2.  Interactions with a light neutral scalar boson can lead to a medium-dependent mass for the neutrino, which will be analyzed.  Various  terrestrial and space-based experimental constraints, as well as astrophysical and cosmological constraints of such scenarios will be presented.