  The observed pattern of fermion masses and mixing is an outstanding puzzle in
particle physics, generally known as the flavor problem. Over the years, guided
by precision neutrino oscillation data, discrete flavor symmetries have often
been used to explain the neutrino mixing parameters, which look very different
from the quark sector. In this review, we discuss the application of
non-Abelian finite groups to the theory of neutrino masses and mixing in the
light of current and future neutrino oscillation data. We start with an
overview of the neutrino mixing parameters, comparing different global fit
results and limits on normal and inverted neutrino mass ordering schemes. Then,
we discuss a general framework for implementing discrete family symmetries to
explain neutrino masses and mixing. We discuss CP violation effects, giving an
update of CP predictions for trimaximal models with nonzero reactor mixing
angle and models with partial $\mu-\tau$ reflection symmetry, and constraining
models with neutrino mass sum rules. The connection between texture zeroes and
discrete symmetries is also discussed. We summarize viable higher-order groups,
which can explain the observed pattern of lepton mixing where the non-zero
$\theta_{13}$ plays an important role. We also review the prospects of
embedding finite discrete symmetries in the Grand Unified Theories and with
extended Higgs fields. Models based on modular symmetry are also briefly
discussed. A major part of the review is dedicated to the phenomenology of
flavor symmetries and possible signatures in the current and future experiments
at the intensity, energy, and cosmic frontiers. In this context, we discuss
flavor symmetry implications for neutrinoless double beta decay, collider
signals, leptogenesis, dark matter, as well as gravitational waves.
