Distributions of transverse momentum p(T)(ll) and the related angular variable phi(eta)*. of Drell-Yan lepton pairs are measured in 20.3 fb(-1) of proton-proton collisions at root s = 8 TeV with the ATLAS detector at the LHC. Measurements in electron-pair and muon-pair final states are corrected for detector effects and combined. Compared to previous measurements in proton-proton collisions at root s = 7 TeV, these new measurements benefit from a larger data sample and improved control of systematic uncertainties. Measurements are performed in bins of lepton-pair mass above, around and below the Z-boson mass peak. The data are compared to predictions from perturbative and resummed QCD calculations. For values of phi(eta)*. < 1 the predictions from the Monte Carlo generator ResBos are generally consistent with the data within the theoretical uncertainties. However, at larger values of phi(eta)*. this is not the case. Monte Carlo generators based on the parton-shower approach are unable to describe the data over the full range of p(T)(ll) while the fixed-order prediction of Dynnlo falls below the data at high values of p(T)(ll). ResBos and the parton-shower Monte Carlo generators provide a much better description of the evolution of the phi(eta)*. and p(T)(ll) distributions as a function of lepton-pair mass and rapidity than the basic shape of the data.