The Cygnus region hosts a giant molecular-cloud complex which actively forms massive stars. Interactions of cosmic rays with interstellar gas and radiation fields make it shine at gamma-ray energies. Several gamma-ray pulsars and other energetic sources are seen in this direction. In this paper we analyse the gamma-ray emission measured by the Fermi Large Area Telescope in the energy range from 100 MeV to 100 GeV in order to probe the gas and cosmic-ray content over the scale of the whole Cygnus complex. The signal from bright pulsars is largely reduced by selecting photons in their off-pulse phase intervals. We compare the diffuse gamma-ray emission with interstellar gas maps derived from radio/mm-wave lines and visual extinction data, and a global model of the region, including other pulsars and gamma-ray sources, is sought. The integral HI emissivity and its spectral energy distribution are both consistent within the systematics with LAT measurements in the interstellar space near the solar system. The average X=N(H2)/W(CO) ratio is consistent with other LAT measurements in the Local Arm. We detect significant gamma-ray emission from dark neutral gas for a mass corresponding to ~40% of that traced by CO. Despite the conspicuous star formation activity and large masses of the interstellar clouds, the cosmic-ray population in the Cygnus complex averaged over a few hundred parsecs is similar to that of the local interstellar space. (abridged)< Réduire