The production of Z bosons with one or two isolated high-energy photons is studied using pp collisions at s=8 TeV. The analyses use a data sample with an integrated luminosity of 20.3 fb-1 collected by the ATLAS detector during the 2012 LHC data taking. The Zγ and Zγγ production cross sections are measured with leptonic (e+e-, μ+μ-, νν) decays of the Z boson, in extended fiducial regions defined in terms of the lepton and photon acceptance. They are then compared to cross-section predictions from the Standard Model, where the sources of the photons are radiation off initial-state quarks and radiative Z-boson decay to charged leptons, and from fragmentation of final-state quarks and gluons into photons. The yields of events with photon transverse energy ET>250 GeV from +-γ events and with ET>400 GeV from ννγ events are used to search for anomalous triple gauge-boson couplings ZZγ and Zγγ. The yields of events with diphoton invariant mass mγγ>200 GeV from +-γγ events and with mγγ>300 GeV from ννγγ events are used to search for anomalous quartic gauge-boson couplings ZZγγ and Zγγγ. No deviations from Standard Model predictions are observed and limits are placed on parameters used to describe anomalous triple and quartic gauge-boson couplings.