This article presents searches for the Zγ decay of the Higgs boson and for narrow high-mass resonances decaying to Z γ, exploiting Z boson decays to pairs of electrons or muons. The data analysis uses 36.1 fb −1 of pp collisions at s = 13 $$ \sqrt{s}=13 $$ recorded by the ATLAS detector at the CERN Large Hadron Collider. The data are found to be consistent with the expected Standard Model background. The observed (expected — assuming Standard Model pp → H → Z γ production and decay) upper limit on the production cross section times the branching ratio for pp → H → Z γ is 6.6. (5.2) times the Standard Model prediction at the 95% confidence level for a Higgs boson mass of 125.09 GeV. In addition, upper limits are set on the production cross section times the branching ratio as a function of the mass of a narrow resonance between 250 GeV and 2.4 TeV, assuming spin-0 resonances produced via gluon-gluon fusion, and spin-2 resonances produced via gluon-gluon or quark-antiquark initial states. For high-mass spin-0 resonances, the observed (expected) limits vary between 88 fb (61 fb) and 2.8 fb (2.7 fb) for the mass range from 250 GeV to 2.4 TeV at the 95% confidence level.