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, $elltoell^prime ubar u$ decays, anomalous magnetic moments of charged leptons, $elltoell^primegamma$ and $ellto3ell^prime$ 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 $tautomu ubar u/tauto e ubar u$ without violating bounds from electroweak precision observables.
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