  We study lepton violating Higgs (HLFV) decays, first from the effective field
theory (EFT) point of view, and then analysing the different high-energy
realizations of the operators of the EFT, highlighting the most promising
models. We argue why two Higgs doublet models can have a
$\mathrm{BR}(h\rightarrow \tau \mu)\sim 0.01$, and why this rate is suppressed
in all other realizations including vector-like leptons. We further discuss
HLFV in the context of neutrino mass models: in most cases it is generated at
one loop giving always $\mathrm{BR}(h\rightarrow \tau \mu) < 10^{-4}$ and
typically much less, which is beyond experimental reach. However, both the Zee
model and extended left-right symmetric models contain extra ${\rm SU(2)}$
doublets coupled to leptons and could in principle account for the observed
excess, with interesting connections between HLFV and neutrino parameters.
