  A search for lepton flavour violating decays of the Higgs boson in the
$\mu\tau$ and e$\tau$ decay modes is presented. The search is based on a data
set corresponding to an integrated luminosity of 35.9 fb$^{-1}$ of
proton-proton collisions collected with the CMS detector in 2016, at a
centre-of-mass energy of 13 TeV. No significant excess over the standard model
expectation is observed. The observed (expected) upper limits on the lepton
flavour violating branching fractions of the Higgs boson are
$\mathcal{B}$(H$\to\mu\tau$) < 0.25% (0.25%) and $\mathcal{B}$(H$\to$e$\tau$) <
0.61% (0.37%), at 95% confidence level. These results are used to derive upper
limits on the off-diagonal $\mu\tau$ and e$\tau$ Yukawa couplings
$\sqrt{|{Y_{\mu\tau}}|^{2}+|{Y_{\tau\mu}}|^{2}} < 1.43 \times 10^{-3}$ and
$\sqrt{|{Y_{\mathrm{e}\tau}}|^{2}+|{Y_{\tau\mathrm{e}}}|^{2}} < 2.26 \times
10^{-3}$ at 95% confidence level. The limits on the lepton flavour violating
branching fractions of the Higgs boson and on the associated Yukawa couplings
are the most stringent to date.
