  Searches for the lepton number violating $K^{+} \rightarrow \pi^{-} \mu^{+}
e^{+}$ decay and the lepton flavour violating $K^{+} \rightarrow \pi^{+}
\mu^{-} e^{+}$ and $\pi^{0} \rightarrow \mu^{-} e^{+}$ decays are reported
using data collected by the NA62 experiment at CERN in $2017$-$2018$. No
evidence for these decays is found and upper limits of the branching ratios are
obtained at 90% confidence level:
$\mathcal{B}(K^{+}\rightarrow\pi^{-}\mu^{+}e^{+})<4.2\times 10^{-11}$,
$\mathcal{B}(K^{+}\rightarrow\pi^{+}\mu^{-}e^{+})<6.6\times10^{-11}$ and
$\mathcal{B}(\pi^{0}\rightarrow\mu^{-}e^{+})<3.2\times 10^{-10}$. These results
improve by one order of magnitude over previous results for these decay modes.
