This paper describes a measurement of the Z/γ∗ boson transverse momentum spectrum using ATLAS proton-proton collision data at a centre-of-mass energy of √s = 7 TeV at the LHC. The measurement is performed in the Z/γ∗ → e⁺e⁻ and Z/γ∗ → μ⁺μ⁻channels, using data corresponding to an integrated luminosity of 4.7 fb⁻¹. Normalized differential cross sections as a function of the Z/γ∗ boson transverse momentum are measured for transverse momenta up to 800 GeV. The measurement is performed inclusively for Z/γ∗ rapidities up to 2.4, as well as in three rapidity bins. The channel results are combined, compared to perturbative and resummed QCD calculations and used to constrain the parton shower parameters of Monte Carlo generators