We present homogeneous abundance determinations for iron and some of the elements involved in the proton-capture reactions (O, Na, Mg, Al, and�Si) for 202�red giants in 17�Galactic globular clusters (GCs) from the analysis of high-resolution UVES spectra obtained with the FLAMES facility at the ESO�VLT2 telescope. Our programme clusters span almost the whole range of the metallicity distribution of GCs and were selected to sample the widest range of global parameters (horizontal-branch morphology, masses, concentration, etc.). In this paper we focus on the discussion of the Na-O and Mg-Al anticorrelations and related issues. Our study finds clear�Na and�O star-to-star abundance variations, exceeding those expected from the error in the analysis, in all clusters. Variations in Al are present in all but a few�GCs. Finally, a spread in abundances of Mg and Si are also present in a few clusters. Mg is slightly less overabundant and Si slightly more overabundant in the most Al-rich stars. The correlation between Si and Al abundances is a signature of production of ²⁸Si leaking from the Mg-Al cycle in a few clusters. The cross sections required for the proper reactions to take over in the cycle point to temperatures in excess of about 65�million K for the favoured site of production. We used a dilution model to infer the total range of Al abundances starting from the Na and Al�abundances in the FLAMES-UVES spectra, and the Na�abundance distributions found from analysis of the much larger set of stars for which FLAMES-GIRAFFE spectra were available. We found that the maximum amount of additional Al produced by first-generation polluters contributing to the composition of the second-generation stars in each cluster is closely correlated with the same combination of metallicity and cluster luminosity that reproduced the minimum O-abundances found from GIRAFFE spectra. We then suggest that the high temperatures required for the Mg-Al cycle are only reached in the most massive and most metal-poor polluters.