A bstract A search for flavour-changing neutral-current processes in top-quark decays is presented. Data collected with the ATLAS detector from proton-proton collisions at the Large Hadron Collider at a centre-of-mass energy of $$ \sqrt{s}=13 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> </mml:math> TeV, corresponding to an integrated luminosity of 36.1 fb −1 , are analysed. The search is performed using top-quark pair events, with one top quark decaying through the t → qZ ( q = u , c ) flavour-changing neutral-current channel, and the other through the dominant Standard Model mode t → bW . Only Z boson decays into charged leptons and leptonic W boson decays are considered as signal. Consequently, the final-state topology is characterized by the presence of three isolated charged leptons (electrons or muons), at least two jets, one of the jets originating from a b -quark, and missing transverse momentum from the undetected neutrino. The data are consistent with Standard Model background contributions, and at 95% confidence level the search sets observed (expected) upper limits of 1.7 × 10 −4 (2.4 × 10 −4 ) on the t → uZ branching ratio and 2.4 × 10 −4 (3.2 × 10 −4 ) on the t → cZ branching ratio, constituting the most stringent limits to date.