  The freeze-in production of Feebly Interacting Massive Particle (FIMP) dark
matter in the early universe is an appealing alternative to the well-known -
and constrained - Weakly Interacting Massive Particle (WIMP) paradigm. Although
challenging, the phenomenology of FIMP dark matter has been receiving growing
attention and is possible in a few scenarios. In this work, we contribute to
this endeavor by considering a $Z^\prime$ portal to fermionic dark matter, with
the $Z^\prime$ having both vector and axial couplings and a mass ranging from
MeV up to PeV. We evaluate the bounds on both freeze-in and freeze-out from
direct detection, atomic parity violation, leptonic anomalous magnetic moments,
neutrino-electron scattering, collider, and beam dump experiments. We show that
FIMPs can already be tested by most of these experiments in a complementary
way, whereas WIMPs are especially viable in the $Z^\prime$ low mass regime, in
addition to the $Z^\prime$ resonance region. We also discuss the role of the
axial couplings of $Z^\prime$ in our results. We therefore hope to motivate
specific realizations of this model in the context of FIMPs, as well as
searches for these elusive dark matter candidates.
