  The ratio $R_K=\frac{\Gamma(K\to e \nu_{e} (\gamma))}{\Gamma(K\to \mu
\nu_{\mu} (\gamma))}$ of leptonic decay rates provides a golden probe for
testing the structure of the weak interactions because it can be predicted with
high accuracy within the Standard Model. The aim of the NA62 experimental
programme based on the 2007 data set is a measurement of $R_K$ reaching a new
accuracy level better than 0.4% \cite{evgueni}. To achieve this goal, data
taking strategy allowing control over the systematic effects and in particular
precise background subtraction, was worked out, and a data sample of
$\sim0.16\times10^6$ $K_{e2}$ candidates with just $\sim10%$ background was
collected. The current status of the $K_{l2}$ analysis based on the dedicated
NA62 data taking is summarized. The achieved precision of background
subtraction, other systematic uncertainties, and prospects of the analysis are
discussed. Within the same scientific programme, NA62, in its second phase,
will measure the branching ratio of the very rare kaon decay $K^+ \to \pi^+ \nu
\overline{\nu}$; the aim is to collect $\mathcal{O}(100)$ events with 10%
background in two years data taking period. The status of the project, the R&D
and future perspectives for the experiment
