Measurements of both the inclusive and differential production cross sections of a top-quark-antiquark pair in association with a Z boson (t (t) over barZ) are presented. The measurements are performed by targeting final states with three or four isolated leptons (electrons or muons) and are based on root s = 13 TeV proton-proton collision data with an integrated luminosity of 139 fb(-1), recorded from 2015 to 2018 with the ATLAS detector at the CERN Large Hadron Collider. The inclusive cross section is measured to be a sigma(t (t) over barZ)= 0.99 +/- 0.05 (stat.) +/- 0.08 (syst.) pb, in agreement with the most precise theoretical predictions. The differential measurements are presented as a function of a number of kinematic variables which probe the kinematics of the t (t) over barZ system. Both absolute and normalised differential crosssection measurements are performed at particle and parton levels for specific fiducial volumes and are compared with theoretical predictions at different levels of precision, based on a chi(2)/ndf and p value computation. Overall, good agreement is observed between the unfolded data and the predictions.