  We studied single $W$ boson production in high energy $e\gamma$ collisions
and the sensitivity of various observables to the $WW\gamma$ gauge boson
coupling. We evaluated the helicity amplitudes including the $W$ decay to final
state fermions and all Feynman diagrams which give the same final state. At
high energy, the non-resonant diagrams give significant contributions to the
cross sections and should not be neglected. We first considered a
$\sqrt{s}=120$~GeV $e^+e^-$ collider converted to an $e\gamma$ collider by
backscattering a low energy laser off of one of the original electron beams.
Such a collider could measure $\kappa_\gamma$ to $\simeq\pm 0.09$ at 95\% C.L.
which is the same level of precision as could be achieved at LEP200 using $W$
pair production. We next considered single $W$ production at a 200 GeV $e^+e^-$
collider in the Weizacker-Williams approximation which can measure the
$WW\gamma$ vertex independently of the $WWZ$ vertex. This process can measure
$\kappa_\gamma$ to $\pm 0.15$ at 95\% C.L. which is comparable to the $W$ pair
production process. Finally, we examined $W$ production at 500 GeV and 1 TeV
$e^+e^-$ colliders, comparing results for photon spectra obtained from a
backscattered laser and from beamstrahlung radiation. Here we found that the
couplings could best be measured using the backscattered laser photons with
$|\delta\kappa_\gamma|\leq 0.07$ and
