A search is presented for resonances decaying to a Z boson and a photon. The analysis is based on data from proton-proton collisions at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb−1, and collected with the CMS detector at the LHC in 2016. Two decay modes of the Z boson are investigated. In the leptonic channels, the Z boson candidates are reconstructed using electron or muon pairs. In the hadronic channels, they are identified using a large-radius jet, containing either light-quark or b quark decay products of the Z boson, via jet substructure and advanced b quark tagging techniques. The results from these channels are combined and interpreted in terms of upper limits on the product of the production cross section and the branching fraction to Zγ for narrow and broad spin-0 resonances with masses between 0.35 and 4.0 TeV, providing thereby the most stringent limits on such resonances.[Figure not available: see fulltext.].