Studies of the spin and parity quantum numbers of the Higgs boson are presented, based on proton-proton collision data collected by the ATLAS experiment at the LHC. The Standard Model spin-parity JP=0+ hypothesis is compared with alternative hypotheses using the Higgs boson decays H→γγ, H→ZZ*→4ℓ and H→WW*→ℓνℓν, as well as the combination of these channels. The analysed dataset corresponds to an integrated luminosity of 20.7 fb-1 collected at a centre-of-mass energy of s=8TeV. For the H→ZZ*→4ℓ decay mode the dataset corresponding to an integrated luminosity of 4.6 fb-1 collected at s=7TeV is included. The data are compatible with the Standard Model JP=0+ quantum numbers for the Higgs boson, whereas all alternative hypotheses studied in this Letter, namely some specific JP=0-, 1+, 1-, 2+ models, are excluded at confidence levels above 97.8%. This exclusion holds independently of the assumptions on the coupling strengths to the Standard Model particles and in the case of the JP=2+ model, of the relative fractions of gluon-fusion and quark-antiquark production of the spin-2 particle. The data thus provide evidence for the spin-0 nature of the Higgs boson, with positive parity being strongly preferred. © 2013 CERN.