  We study the anomalous $ZZ\gamma$ and $Z\gamma\gamma$ couplings that can be
probed via $Z\gamma$ production at the $e^+ \, e^-$ colliders. We take Standard
Model Effective Field Theory (SMEFT) approach to examine these anomalous
neutral triple gauge couplings in a model independent way. There are four
independent dimension-8 operators that generate these gauge interactions, one
of them is CP-conserving and rest three are CP-violating. We adopt optimal
observable technique to extract the sensitivity at which these anomalous
couplings can be probed at future $e^+e^-$ colliders and then compare the
results with the latest experimental limit obtained at the LHC. We also study
the impact and advantage of beam polarization in these precision measurements.
Statistical limit (95\% C.L.) on individual anomalous couplings as well as the
correlation between various couplings have been discussed in detail.
