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The NA62 experiment at the CERN SPS is designed to measure the branching ratio of the K+→π+vv decay, very precisely predicted below 10exp[-10] by the SM. NA62 took data in 2016-2018 and collected a large sample of charged kaon decays into final states with multiple charged particles. 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.