We review standard and non-standard neutrino physics probes that are based on nuclear measurements. We pay special attention on the discussion of prospects to extract new physics at prominent rare event measurements looking for neutrino-nucleus scattering, such as the coherent elastic neutrino-nucleus scattering (CE$ u$NS) that may involve lepton flavor violation (LFV) in neutral-currents (NC). For the latter processes several appreciably sensitive experiments are currently pursued or have been planed to operate in the near future, like the COHERENT, CONUS, CONNIE, MINER, TEXONO, RED100, vGEN, Ricochet, NUCLEUS etc. We provide a thorough discussion on phenomenological and theoretical studies, in particular those referring to the nuclear physics aspects in order to provide accurate predictions for the relevant experiments. Motivated by the recent discovery of CE$ u$NS at the COHERENT experiment and the active experimental efforts for a new measurement at reactor-based experiments, we summarize the current status of the constraints as well as the future sensitivities on nuclear and electroweak physics parameters, non-standard interactions, electromagnetic neutrino properties, sterile neutrinos and simplified scenarios with novel vector $Z^prime$ or scalar $phi$ mediators. Indirect and direct connections of cevns with astrophysics, direct Dark Matter detection and charge lepton flavor violating processes are also discussed.
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