  The status of tests of the standard electroweak model and of searches for new
physics in allowed nuclear $\beta$ decay and neutron decay is reviewed
including both theoretical and experimental developments. The sensitivity and
complementarity of recent and ongoing experiments are discussed with emphasis
on their potential to look for new physics. Measurements are interpreted using
a model-independent effective field theory approach enabling to recast the
outcome of the analysis in many specific new physics models. Special attention
is given to the connection that this approach establishes with high-energy
physics. A new global fit of available $\beta$-decay data is performed
incorporating, for the first time in a consistent way, superallowed $0^+\to
0^+$ transitions, neutron decay and nuclear decays. The constraints on exotic
scalar and tensor couplings involving left- or right-handed neutrinos are
determined while a constraint on the pseudoscalar coupling from neutron decay
data is obtained for the first time as well. The values of the vector and
axial-vector couplings, which are associated within the standard model to
$V_{ud}$ and $g_A$ respectively, are also updated. The ratio between the axial
and vector couplings obtained from the fit under standard model assumptions is
$C_A/C_V = -1.27510(66)$. The relevance of the various experimental inputs and
error sources is critically discussed and the impact of ongoing measurements is
studied. The complementarity of the obtained bounds with other low- and
high-energy probes is presented including ongoing searches at the Large Hadron
Collider.
