Physics beyond the SM with kaons at NA62

The decay $K^{+} \rightarrow \pi^{+} v \overline{v}$, with a very precisely predicted branching ratio of less than $10^{-10}$, is one of the best candidates to reveal indirect effects of new physics at the highest mass scales. The NA62 experiment at the CERN SPS is designed to measure the branching ratio of the $K^{+} \rightarrow \pi^{+} v \overline{v}$ with a decay-in-flight technique. NA62 took data so far in 2016-2018. Statistics collected in 2016 allowed NA62 to reach the Standard Model sensitivity for $K^{+} \rightarrow \pi^{+} v \overline{v}$, entering the domain of $10^{-10}$ single event sensitivity and showing the proof of principle of the experiment. Thanks to the statistics collected in 2017, NA62 surpasses the present best sensitivity. The analysis strategy is reviewed and the preliminary result from the 2017 data set is presented.The NA62 experiment at CERN collected a large sample of charged kaon decays into final states with multiple charged particles in 2016-2018. The sensitivity to a range of lepton flavour and lepton number violating kaon decays provided by this data set improves over the previously reported measurements. Results from the searches for these processes with a partial NA62 data sample are presented.