Abstract : The gamma-ray sky >100 MeV is dominated by the diffuse emissions from interactions of cosmic rays with the interstellar gas and radiation fields of the Milky Way. Observations of these diffuse emissions provide a tool to study cosmic-ray origin and propagation, and the interstellar medium. We present measurements from the first 21 months of the Fermi-LAT mission and compare with models of the diffuse gamma-ray emission generated using the GALPROP code. The models are fitted to cosmic-ray data and incorporate astrophysical input for the distribution of cosmic-ray sources, interstellar gas and radiation fields. To assess uncertainties associated with the astrophysical input, a grid of models is created by varying within observational limits the distribution of cosmic-ray sources, the size of the cosmic-ray confinement volume (halo), and the distribution of interstellar gas. An all-sky maximum-likelihood fit is used to determine the Xco-factor, the ratio between integrated CO-line intensity and molecular hydrogen column density, the fluxes and spectra of the gamma-ray point sources from the first Fermi-LAT catalogue, and the intensity and spectrum of the isotropic background including residual cosmic rays that were misclassified as gamma rays, all of which have some dependency on the assumed diffuse emission model. The models are compared on the basis of their maximum likelihood ratios as well as spectra, longitude, and latitude profiles. We also provide residual maps for the data following subtraction of the diffuse emission models. The models are consistent with the data at high and intermediate latitudes but under-predict the data in the inner Galaxy for energies above a few GeV. Possible explanations for this discrepancy are discussed, including the contribution by undetected point source populations and spectral variations of cosmic rays throughout the Galaxy.