  We speculate about the possible interpretations of the recently observed
excess in the $h \to \tau \mu$ decay. We derive a robust lower bound on the
Higgs boson coupling strength to a tau and a muon, even in presence of the most
general new physics affecting other Higgs properties. Then we reevaluate
complementary indirect constraints coming from low energy observables as well
as from theoretical considerations. In particular, the tentative signal should
lead to $\tau \to \mu \gamma$ at rates which could be observed at Belle II. In
turn we show that, barring fine-tuned cancellations, the effect can only be
accommodated within models with an extended scalar sector. These general
conclusions are demonstrated using a number of explicit new physics models.
Finally we show how, given the $h \to \tau \mu$ signal, the current and future
searches for $\mu \to e \gamma$ and $\mu \to e$ nuclear conversions
unambiguously constrain the allowed rates for $h \to \tau e$.
