The first direct search for lepton-flavour-violating decays of the recently discovered Higgs boson (H) is described. The search is performed in the H -> mu tau(e) and H -> mu tau(h) channels, where tau(e) and 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 root 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-H = 125 GeV is 0.010. The best fit branching fraction is beta(H -> mu tau) = (0.84(-0.37)(+0.39))%. A constraint on the branching fraction, beta(H -> mu tau) < 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 x 10(-3). (C) 2015 CERN for the benefit of the CMS Collaboration. Published by Elsevier B.V.