  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$\nu$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$\nu$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.
