  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. [Abridged]
