  Whilst the prospect of new $Z'$ gauge bosons with only axial couplings to the
Standard Model (SM) fermions is widely discussed, examples of anomaly-free
renormalisable models are lacking in the literature. We look to remedy this by
constructing several motivated examples. Specifically, we consider axial
vectors which couple universally to all SM fermions, as well as those which are
generation-specific, leptophilic, and leptophobic. Anomaly cancellation
typically requires the presence of new coloured and charged chiral fermions,
and we argue that in a large class of models masses of these new states are
expected to be comparable to that of the axial vector. Finally, an axial vector
mediator could provide a portal between SM and hidden sector states, and we
also consider the possibility that the axial vector couples to dark matter. If
the dark matter relic density is set due to freeze-out via the axial vector,
this strongly constrains the parameter space.
