"A unique and innovative test of a cornerstone principle of the Standard Model of particle physics, the Lepton Favour (LF) conservation, is proposed in the framework of the NA62 experiment at CERN. The search for nine decay modes of the charged kaon and the neutral pion forbidden in the Standard Model by LF conservation will be carried out at a record sensitivity of one part in a trillion. Such sensitivity will be achieved due to the uniquely intense kaon beam that will become available to the experiment in 2014, as well as a range of novel particle detection technologies employed. The collection of the LF violating decay candidates will take place in ""parasitic"" mode alongside main NA62 data taking, which guarantees the feasibility, high data quality and cost-effectiveness. The project will bridge a significant research gap that has developed due to the absence of dedicated LF projects in the kaon sector, in sharp contrast with B-meson, lepton and neutrinoless double beta decay experiments. Any observed LF violating process will unambiguously point to physical phenomena beyond the Standard Model description, and will thus represent a major discovery. The Standard Model extensions that will be probed include those involving heavy Majorana neutrinos and R-parity breaking supersymmetry. Entire classes of new physics models will be confirmed, rigorously constrained or eliminated."