  We study high-energy $\gamma$-rays observed from Cyg X-3 by the Fermi Large
Area Telescope and the 15-GHz emission observed by the Ryle Telescope and the
Arcminute Microkelvin Imager. We measure the $\gamma$-ray spectrum averaged
over strong flares much more accurately than before, and find it well modelled
by Compton scattering of stellar radiation by relativistic electrons with the
power law index of $\simeq$3.5 and a low-energy cutoff at the Lorentz factor of
$\sim\!10^3$. We find a weaker spectrum in the soft spectral state, but only
upper limits in the hard and intermediate states. We measure strong orbital
modulation during the flaring state, well modelled by anisotropic Compton
scattering of blackbody photons from the donor by jet relativistic electrons.
We discover a weaker orbital modulation of the 15 GHz radio emission, which is
well modelled by free-free absorption by the stellar wind. We then study
cross-correlations between radio, $\gamma$-ray and X-ray emissions. We find the
cross-correlation between the radio and $\gamma$-ray emissions peaks at a lag
less than 1 d, while we detect a distinct radio lag of $\sim$50 d with respect
to the soft X-rays in the soft spectral state.
