What have the Planck data taught us about inflation?

International audienceWe discuss the implications of the recently released Planck data for inflation. We show that they favor the simplest category of inflationary models, namely single-field slow-roll inflation, with a plateau-like potential, a typical representative of this class of scenarios being the Starobinsky model. We also investigate the constraints on the inflationary reheating. Finally, we briefly address the question of which observables should be measured as a matter of priority in order to further improve our knowledge of inflation