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, (l(+)l(-)l(+)l(-) and l(+)l(-)v(v) over bar), quark and lepton pairs, (q(q) over bar l)+l-, (q(q) over bar v(v) over bar) and only hadrons (q(q) over bar q(q) over bar) are considered. In all states with at least one Z boson decaying hadronically, lifetime, lepton and event-shape tags are used to separate b(b) over bar pairs from q(q) over bar final states. Limits on anomalous ZZgamma and ZZZ couplings are derived from the measured cross sections and from event kinematics using an optimal observable method. Limits on low scale gravity with large extra dimensions are derived from the cross sections and their dependence on polar angle.