  Current neutrino experiments measure the neutrino mixing parameters with an
unprecedented accuracy. The upcoming generation of experiments will be
sensitive to subdominant effects that can give information on the unknown
neutrino parameters: the Dirac CP-violating phase, the mass ordering and the
$\theta_{23}$ octant. Determining the exact values of neutrino mass and mixing
parameters is crucial to test neutrino models and flavor symmetries. In the
first part of this review, we summarize the current status of neutrino
oscillation parameters. We consider the most recent data from solar experiments
and the atmospheric data from Super-Kamiokande, IceCube and ANTARES. We
implement the data from the reactor experiments KamLAND, Daya Bay, RENO and
Double Chooz as well as the long baseline data from MINOS, T2K and NOvA. If in
addition to the standard interactions, neutrinos have subdominant Non-Standard
Interactions (NSI) with matter, extracting the values of these parameters will
suffer from new degeneracies. We review such effects and formulate the
conditions on the NSI parameters under which the precision measurement of
neutrino oscillation parameters can be distorted. Like standard weak
interactions, NSI can be categorized into Charged and Neutral Current NSI. Our
focus will be on NC NSI since it is possible to build a class of models giving
rise to sizeable NC NSI with effects on neutrino oscillations. These models are
based on new $U(1)$ gauge symmetry with a boson of mass $\lesssim 10$ MeV. The
UV complete model should be electroweak invariant which implies that along with
neutrinos, charged fermions acquire new interactions on which there are strong
bounds. We enumerate the bounds that exist on such models and show that it is
possible to build viable models avoiding all the bounds. We review methods to
test these models and suggest approaches to break the degeneracies caused by
NSI.
