For more than one year the Fermi Large Area Telescope has been surveying the
gamma-ray sky from 20 MeV to more than 300 GeV with unprecedented statistics
and angular resolution. One of the key science targets of the Fermi mission is
diffuse gamma-ray emission. Galactic interstellar gamma-ray emission is
produced by interactions of high-energy cosmic rays with the interstellar gas
and radiation field. We review the most important results on the subject
obtained so far: the non-confirmation of the excess of diffuse GeV emission
seen by EGRET, the measurement of the gamma-ray emissivity spectrum of local
interstellar gas, the study of the gradient of cosmic-ray densities and of the
X(CO)=N(H2)/W(CO) ratio in the outer Galaxy. We also catch a glimpse at diffuse
gamma-ray emission in the Large Magellanic Cloud. These results allow the
improvement of large-scale models of Galactic diffuse gamma-ray emission and
new measurements of the extragalactic gamma-ray background.Comment: Contribution to the Workshop SciNeGHe 2009/Gamma-ray Physics in the
LHC era (Assisi - Italy, Oct. 7-9 2009); 10 pages, 6 figure