Abstract: A search for flavour-changing neutral current decays of a top quark into an up-type quark (q = u, c) and the Standard Model Higgs boson, t Hq, is presented. The search is based on a dataset of pp collisions at = 13 TeV recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider and corresponding to an integrated luminosity of 36.1 fb(-1). Two complementary analyses are performed to search for top-quark pair events in which one top quark decays into Wb and the other top quark decays into Hq, and target the Hbb and H (+-) decay modes, respectively. The high multiplicity of b-quark jets, or the presence of hadronically decaying -leptons, is exploited in the two analyses respectively. Multivariate techniques are used to separate the signal from the background, which is dominated by top-quark pair production. No significant excess of events above the background expectation is found, and 95% CL upper limits on the t Hq branching ratios are derived. The combination of these searches with ATLAS searches in diphoton and multilepton final states yields observed (expected) 95% CL upper limits on the t Hc and t Hu branching ratios of 1.1 x 10(-3) (8.3 x 10(-4)) and 1.2 x 10(-3) (8.3 x 10(-4)), respectively. The corresponding combined observed (expected) upper limits on the |(tcH)| and |(tuH)| couplings are 0.064 (0.055) and 0.066 (0.055), respectively.