Measurements of the production of jets of particles in association with a Z boson in pp collisions at root s = 7 TeV are presented, using data corresponding to an integrated luminosity of 4.6 fb(-1) collected by the ATLAS experiment at the Large Hadron Collider. Inclusive and differential jet cross sections in Z events, with Z decaying into electron or muon pairs, are measured for jets with transverse momentum p(T) > 30 GeV and rapidity vertical bar y vertical bar < 4.4. The results are compared to next-to-leading-order perturbative QCD calculations, and to predictions from different Monte Carlo generators based on leading-order and next-to-leading-order matrix elements supplemented by parton showers.