The correlation between the spectral peak photon energy, <em>E</em>p, and the radiated energy or luminosity (i.e., the “Amati relation” and other correlations derived from it) is one of the central and most debated topics in GRB astrophysics, with implications for physics and the geometry of prompt emission, the identification and understanding of various classes of GRBs (short/long, XRFs,sub-energetic), and GRB cosmology. Fermi is exceptionally suited to provide, also in conjunction with Swift observations, a significant step forward in this field of research. Indeed, one of the main goals of Fermi/GBM is to make accurate measurements of <em>E</em>p, by exploiting its unprecedented broad energy band from ~8 keV to ~30MeV; in addition, for a small fraction of GRBs, the LAT can extend the spectral measurements up to the GeV energy range, thus allowing a reliable estimate of the bolometric radiated energy/luminosity. We provide a review, an update and a discussion of the impact of Fermi observations in the investigation, understanding and testing of the <em>E</em>_p,i –<em>E</em>_iso (“Amati”) relation.