  Several experiments reported hints for the violation of lepton flavor or
lepton flavor universality in processes involving muons. Most prominently,
there is the hint for a non-zero rate of the flavor violating Higgs decay $h
\to \tau\mu$ at the LHC, as well as the hint for lepton flavor universality
violation in rare $B$ meson decays at LHCb. In addition, also the long standing
discrepancy in the anomalous magnetic moment of the muon motivates new physics
connected to muons. A symmetry which violates lepton flavor universality, is
$L_\mu-L_\tau$: the difference of muon-number and tau-number. We show that
adding vector-like fermions to a $L_\mu-L_\tau$ theory generates naturally an
effect in the anomalous magnetic moment of the muon and $h\to\tau\mu$, while
effects in other $\tau \to \mu$ transitions are systematically suppressed by
symmetry arguments. We find that if $L_\mu-L_\tau$ is gauged it is possible to
also accommodate the discrepant $b\to s\mu\mu$ data while predicting a
$\tau\to3\mu$ and a modified $h\to\mu\mu$ rate within reach of upcoming
experiments.
