This Letter presents direct searches for lepton flavour violation in Higgs boson decays, H -> e tau and H -> mu tau, performed with the ATLAS detector at the LHC. The searches are based on a data sample of proton-proton collisions at a centre-of-mass energy root s = 13 TeV, corresponding to an integrated luminosity of 36.1 fb(-1). No significant excess is observed above the expected background from Standard Model processes. The observed (median expected) 95% confidence-level upper limits on the lepton-flavour-violating branching ratios are 0.47%(0.34(-0.10)(+0.13)%) and 0.28%(0.37(-0.10)(+0.14)%) for H -> e tau and H -> mu tau, respectively. (C) 2019 The Author(s). Published by Elsevier B.V.