The NA62 experiment at CERN has collected a large sample of $K^+$ decays in flight during several data-taking periods in $2016 - 2017 - 2018$ (Run1) and the ongoing Run2, which started in 2021. Dedicated lepton-pair trigger lines have enabled a broad rare and forbidden decay programme. A review of searches for lepton flavour and number violating $K^+$ and $\pi^0$ decays with the NA62 Run1 lepton-pair dataset is presented. Ten final states have been investigated, improving on the world data in each case. Upper limits of the decay branching fractions are obtained typically at the $10^{-11}$ level, and the sensitivities are not limited by backgrounds. Prospects of the searches with the full NA62 dataset are also discussed.