  Decays of the neutral and long-lived $\eta$ and $\eta'$ mesons provide a
unique, flavor-conserving laboratory to test low-energy Quantum Chromodynamics
and search for new physics beyond the Standard Model. They have drawn
world-wide attention in recent years and have inspired broad experimental
programs in different high-intensity-frontier centers. New experimental data
will offer critical inputs to precisely determine the light quark mass ratios,
$\eta$-$\eta'$ mixing parameters, and hadronic contributions to the anomalous
magnetic moment of the muon. At the same time, it will provide a sensitive
probe to test potential new physics. This includes searches for hidden photons,
light Higgs scalars, and axion-like particles that are complementary to
worldwide efforts to detect new light particles below the GeV mass scale, as
well as tests of discrete symmetry violation. In this review, we give an update
on theoretical developments, discuss the experimental opportunities, and
identify future research needed in this field.
