We have obtained measurements and upper limits on the emission of Cyg X-1 in the photon energy range of 0.03-300 GeV based on observations by Fermi. We present the results separately for the hard and soft spectral states, as well as for all of the analysed data. In the hard state, we detect a weak steady emission in the 0.1-10 GeV range with a power-law photon index of γ {minus tilde} 2.6 ± 0.2 at a 4σ statistical significance. This measurement, even if considered to be an upper limit, strongly constrains Compton emission of the steady radio jet, present in that state. The number of relativistic electrons in the jet has to be low enough for the spectral components due to Compton upscattering of the stellar blackbody and synchrotron radiation to be within the observed fluxes. If optically thin synchrotron emission of the jet is to account for the MeV tail, as implied by the recently claimed strong polarization in that energy range, the magnetic field in the jet has to be much above equipartition. The GeV-range measurements also strongly constrain models of hot accretion flows, most likely present in the hard state, in which γ -rays are produced from decay of neutral pions produced in collisions of energetic ions in an inner part of the flow. In the soft state, the obtained upper limits constrain electron acceleration in a non-thermal corona, most likely present around a blackbody accretion disc. The coronal emission above 30 MeV has to be rather weak, which is most readily explained by absorption of γ -rays in pair-producing photon-photon collisions. Then, the size of the bulk of the corona is less than a few tens of the gravitational radii. © 2013 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.