A study of Z-boson pair production 
in $\mathrm{ e^+ e^-}$ annihilation at center-of-mass
energies between
$190$\,GeV and  $209$\,GeV is reported.
Final states containing only leptons,
($\ell \ell \ell \ell$ and $\ell \ell \nu \nu$),
quark and lepton pairs,
($\mathrm{qq \ell \ell}$, $\mathrm{qq \nu \nu}$)
and the all-hadronic final state 
($\mathrm{qqqq}$) are considered.
In all states with at least one Z boson decaying
hadronically, $\mathrm{ q \bar{q}}$ and $\mathrm{ b \bar{b}}$ 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.