A measurement of production cross sections of the Higgs boson in proton-proton collisions is presented in the H → τ τ decay channel. The analysis is performed using 36.1 fb -1 of data recorded by the ATLAS experiment at the Large Hadron Collider at a center-of-mass energy of √ s = 13 TeV. All combinations of leptonic ( τ → l v v ‾ with l = e , μ ) and hadronic ( τ → hadrons v ) τ decays arc considered. The H → τ τ signal over the expected background from other Standard Model processes is established with an observed (expected) significance of 4.4 (4.1) standard deviations. Combined with results obtained using data taken at 7 and 8 TeV center-of-mass energies, the observed (expected) significance amounts to 6.4 (5.4) standard deviations and constitutes an observation of H → τ τ decays. Using the data taken at √ s = 13 TeV, the total cross section in the H → τ τ decay channel is measured to be 3.77 +0.60 -0.59 (stat) +0.87 -0.74 (syst) pb, for a Higgs boson of mass 125 GeV assuming the relative contributions of its production modes as predicted by the Standard Model. Total cross sections in the H → τ τ decay channel are determined separately for vector-boson-fusion production and gluon-gluon-fusion production to be σ VBF H → τ τ = 0.28 ± 0.09 (stat) +0.11 -0.09 (syst) pb and σ ggF H → τ τ = 3.1 ± 1.0 (stat) +1.6 -1.3 (syst) pb, respectively. Similarly, results of a fit are reported in the framework of simplified template cross sections. All measurements are in agreement with Standard Model expectations.