  New neutral heavy gauge bosons ($Z^\prime$) are predicted within many
extensions of the Standard Model. While in case they couple to quarks the LHC
bounds are very stringent, leptophilic $Z^\prime$ bosons (even with sizable
couplings) can be much lighter and therefore lead to interesting quantum
effects in precision observables (like $(g-2)_\mu$) and generate flavour
violating decays of charged leptons. In particular,
$\ell\to\ell^\prime\nu\bar\nu$ decays, anomalous magnetic moments of charged
leptons, $\ell\to\ell^\prime\gamma$ and $\ell\to3\ell^\prime$ decays place
stringent limits on leptophilic $Z^\prime$ bosons. Furthermore, in case of
mixing $Z^\prime$ with the SM $Z$, $Z$ pole observables are affected. In light
of these many observables we perform a global fit to leptophilic $Z^\prime$
models with the main goal of finding the bounds for the $Z^\prime$ couplings to
leptons. To this end we consider a number of scenarios for these couplings.
While in generic scenarios correlations are weak, this changes once additional
constraints on the couplings are imposed. In particular, if one considers an
$L_\mu-L_\tau$ symmetry broken only by left-handed rotations, or considers the
case of $\tau-\mu$ couplings only. In the latter setup, on can explain the
$(g-2)_\mu$ anomaly and the hint for lepton flavour universality violation in
$\tau\to\mu\nu\bar\nu/\tau\to e\nu\bar\nu$ without violating bounds from
electroweak precision observables.
