We present the results of the spectral analysis of the public data of 438 Gamma Ray Bursts (GRBs) detected by the Fermi Gamma ray Burst Monitor (GBM) up to March 2010. For 432 bursts we could fit the time integrated spectrum. In 318 cases we can reliably constrain the peak energy Epeak of their \nu F_\nu spectrum by analyzing their time integrated spectrum between 8 keV and 35 MeV. 80% of these spectra are fitted by a power law with an exponential cutoff, and the remaining with the Band function. Among these 318 GRBs, 274 and 44 belong to the long and short GRB class, respectively. Long GRBs have a typical peak energy Epeak=160 keV and low energy spectral index alpha=-0.92. Short GRBs have harder peak energy (Epeak=490 keV) and harder low energy spectral index (alpha=-0.50) than long bursts. For each Fermi GRB we analyzed also the spectrum corresponding to the peak flux of the burst. On average, the peak spectrum has harder low energy spectral index but similar Epeak than the corresponding time-integrated spectrum for the same burst. The spectral parameters derived in our analysis of Fermi/GBM bursts are globally consistent with those reported in the GRB Cicular Network (GCN) archive after December 2008, while we found systematic differences, concerning the low energy power law index, for earlier bursts.