Precision SM calculations and theoretical interests beyond the SM in K_(l2) and K_(l3) decays

We present a brief overview of the recent theoretical progress to describe the K_{l2} and K_{l3} decays. We discuss the interesting probes of the Standard Model offered by these decays such as the extraction of |V_{us}| and the test of the CKM unitarity, the tests of lepton universality, the quark mass ratio determination and the test of the electroweak couplings of the light quarks to the W-boson.