The first direct search for lepton-flavour-violating decays of the recently discovered Higgs boson (H) is described. The search is performed in the $\mathrm{H \to \mu \tau_e}$ and $\mathrm{H \to \mu \tau_h}$ channels, where $\mathrm{\tau_e}$ and $\mathrm{\tau_h}$ are tau leptons reconstructed in the electronic and hadronic decay channels, respectively. The data sample used in this search was collected in pp collisions at a centre-of-mass energy of $\sqrt{s}$ = 8 TeV with the CMS experiment at the CERN LHC and corresponds to an integrated luminosity of 19.7 fb$^{-1}$. The sensitivity of the search is an order of magnitude better than the existing indirect limits. A slight excess of signal events with a significance of 2.4 standard deviations is observed. The $p$-value of this excess at $M_{\mathrm{H}}$ = 125 GeV is 0.010. The best fit branching fraction is $\mathcal{B}(\mathrm{H \to \mu \tau} )=(0.84^{+0.39}_{-0.37})$%. A constraint on the branching fraction, $\mathcal{B}(\mathrm{H \to \mu \tau})$ less than 1.51% at 95% confidence level, is set. This limit is subsequently used to constrain the $\mu$-$\tau$ Yukawa couplings to be less than $3.6\times 10^{-3}$.

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