Searches for lepton flavour and lepton number violation in K$^{\mathrm{+}}$ decays

The NA62 experiment at CERN collected a large sample of charged kaon decays into final states with multiple charged particles in 2016-2018. This sample provides sensitivities to rare decays with branching ratios as low as 10$^{\mathrm{-11}}$. Based on this data set, searches for lepton flavour and lepton number violating decays of the charged kaon into final states containing a lepton pair are presented.