A bstract Charged Higgs bosons produced either in top-quark decays or in association with a top-quark, subsequently decaying via H ± → τ ± ν τ , are searched for in 36.1 fb −1 of proton-proton collision data at $$ \sqrt{s}=13 $$ <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 recorded with the ATLAS detector. Depending on whether the top-quark produced together with H ± decays hadronically or leptonically, the search targets τ +jets and τ +lepton final states, in both cases with a hadronically decaying τ -lepton. No evidence of a charged Higgs boson is found. For the mass range of $$ {m}_{H^{\pm }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>±</mml:mo> </mml:msup> </mml:msub> </mml:math> = 90–2000 GeV, upper limits at the 95% confidence level are set on the production cross-section of the charged Higgs boson times the branching fraction $$ \mathrm{\mathcal{B}}\left({H}^{\pm}\to {\tau}^{\pm }{\nu}_{\tau}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℬ</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> in the range 4.2–0.0025 pb. In the mass range 90–160 GeV, assuming the Standard Model cross-section for $$ t\overline{t} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> <mml:mover> <mml:mi>t</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> production, this corresponds to upper limits between 0.25% and 0.031% for the branching fraction $$ \mathrm{\mathcal{B}}\left(t\to b{H}^{\pm}\right)\times \mathrm{\mathcal{B}}\left({H}^{\pm}\to {\tau}^{\pm }{\nu}_{\tau}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℬ</mml:mi> <mml:mfenced> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mi>b</mml:mi> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>±</mml:mo> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mo>×</mml:mo> <mml:mi>ℬ</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> .