  The $\eta$ and $\eta^{\prime}$ mesons are nearly unique in the particle
universe since they are almost Goldstone bosons and the dynamics of their
decays are strongly constrained. The integrated $\eta$-meson samples collected
in earlier experiments amount to $\sim10^{9}$ events. A new experiment, REDTOP
(Rare Eta Decays To Probe New Physics), is being proposed, with the intent of
collecting a data sample of order 10$^{14}$ $\eta$ (10$^{12}$ $\eta^{\prime}$)
for studying very rare decays. Such statistics are sufficient for investigating
several symmetry violations, and for searching for particles and fields beyond
the Standard Model. In this work we present several studies evaluating REDTOP
sensitivity to processes that couple the Standard Model to New Physics through
all four of the so-called \emph{portals}: the Vector, the Scalar, the Axion and
the Heavy Lepton portal. The sensitivity of the experiment is also adequate for
probing several conservation laws, in particular $CP$, $T$ and Lepton
Universality, and for the determination of the $\eta$ form factors, which is
crucial for the interpretation of the recent measurement of muon $g-2$.
