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->gamma gamma, H -> ZZ -> 4 leptons and H->WW -> l nu l nu, 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 sqrt(s) = 8 TeV. For the H -> ZZ -> 4-lepton decay mode the dataset corresponding to an integrated luminosity of 4.6 fb-1 collected at sqrt(s) = 7 TeV is added. 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.