A study of Z-boson pair production in e^+e^- annihilation at center-of-mass energies between 190 GeV and 209 GeV is reported. Final states containing only leptons, (llll and ll nu nu), quark and lepton pairs, (qqll, qq nu nu) and only hadrons (qqqq) are considered. In all states with at least one Z boson decaying hadronically, qqbar and bbbar final states are considered separately using lifetime, lepton and event-shape tags. Limits on anomalous ZZ gamma and ZZZ couplings are derived from the measured cross section and from event kinematics using an optimal observable method. Limits on low scale gravity with extra dimensions are derived from the cross section and their dependence on polar angle.