  The Gamma-ray Burst Monitor (GBM) on board Fermi allows to study the spectra
of Gamma Ray Bursts (GRBs) over an unprecedented wide energy range (8 keV - 35
MeV). We compare the spectral properties of short and long GRBs detected by the
GBM (up to March 2010) with those of GRBs detected by the BATSE instrument on
board the CGRO. GBM and BATSE long bursts have similar distributions of fluence
(F), Epeak and peak flux (P) but GBM bursts have a slightly harder low-energy
spectral index \alpha with respect to BATSE GRBs. GBM and BATSE short bursts
have similar distributions of fluence, \alpha and peak flux, with GBM bursts
having slightly larger Epeak. We discuss these properties in light of the found
correlations between Epeak and the fluence and the peak flux. GBM bursts
confirm that these correlations are not determined by instrumental selection
effects. Indeed, GBM bursts extend the Epeak-Fluence and Epeak-P correlations
both in fluence/peak flux and in peak energy. No GBM long burst with Epeak
exceeding a few MeV is found, despite the possibility of detecting it.
Similarly to what found with BATSE, there are 3% of GBM long bursts (and almost
all short ones) that are outliers at more than 3\sigma of the Epeak-Eiso
correlation. Instead there is no outlier of the Epeak-Lp,iso correlation, for
both long and short GBM bursts.
