  We study the process $e^+e^- \rightarrow W^+W^- \gamma$ at high-energy $e^+
e^-$ colliders to investigate the effect of genuine quartic
$W^+W^-\gamma\gamma$ and $W^+W^- Z\gamma$ anomalous couplings on the cross
section. Deviations from the Standard Model predictions are quantified. We show
how bounds on the anomalous couplings can be improved by choosing specific
initial state helicity combinations. The dependence of the anomalous
contributions on the collider energy is studied.
