  The discovery of a new neutral gauge boson, $Z'$, could provide the first
concrete evidence of physics beyond the standard model. We explore how nuclear
weak charge measurements in atomic parity violation (APV) experiments can be
used to constrain $Z'$ bosons. We use the recent measurement of the $^{133}$Cs
nuclear weak charge to estimate lower bounds on the mass of $Z'$ bosons for a
number of representative models and to put constraints on the couplings of a
newly discovered $Z'$ boson. We also consider how these constraints might be
improved by future APV experiments that will measure nuclear weak charges of
multiple isotopes. We show how measurements of a single isotope, and combining
measurements into ratios and differences, can be used to constrain the
couplings of a $Z'$ and discriminate between models. We find that current and
future APV experiments could potentially play an important role in unravelling
new physics if a $Z'$ were discovered.
