K Rare Decays with NA62

The long-term goal of NA62 is to measure the ultra rare K+/- -> pi+/- nu anti-nu decay with a sensitivity of 10^-12 per event. This is done by using the decay - in - flight technique which allows a signal acceptance of ~10% . The aim is to collect about 100 signal events in two years of data taking with a background to signal ratio smaller than 10%. The principle of the experimental measurement and the layout of the detector are presented. During 2007/2008 a dedicated run devoted to NA62 prototype tests and study of K_e2 decays was taken. The first phase of the NA62 experiment is aiming at a high precision test of the lepton universality by measuring the helicity suppressed ratio R_K. The preliminary result based on 40% of the 2007 NA62 data sample, R_K = K_e2/K_mu2 = (2.500 +/- 0.016) x 10^-5, which is the first result with a precision better than 1%, is consistent with the Standard Model. Aiming at charge asymmetry measurements, the NA48/2 experiment collected an unprecedented amount of charged K_(3 pi) events. The large samples allowed a precision measurement of rare charged kaon decays. New measurements of the K+/-_(pi l l) decays based on the full NA48/2 data sample collected during 2003/2004 are reported in this paper. Samples of about 7200 reconstructed K+/- -> pi+/- e+ e- events, and more than 3000 K+/- -> pi+/- mu+ mu- events, with a few percent background contamination, have been collected. A precise measurement of th e branching fractions and the form factors of the rare decays K+/-_(pi l l)$ were performed. Measurements of the CP-violating and the forward-backward asymmetries are reported.