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We stress the importance of precise measurements of rare decays $K^+\rightarrowπ^+ν\barν$, $K_L\rightarrowπ^0ν\barν$, $K_{L,S}\toμ^+μ^-$ and $K_{L,S}\toπ^0\ell^+\ell^-$ for the search of new physics (NP). This includes both branching ratios and the distributions in $q^2$, the invariant mass-squared of the neutrino system in the case of $K^+\rightarrowπ^+ν\barν$ and $K_L\rightarrowπ^0ν\barν$ and of the $\ell^+\ell^-$ system in the case of the remaining decays. In particular the correlations between these observables and their correlations with the ratio $\varepsilon'/\varepsilon$ in $K_L\toππ$ decays, the CP-violating parameter $\varepsilon_K$ and the $K^0-\bar K^0$ mass difference $ΔM_K$, should help to disentangle the nature of possible NP. We stress the strong sensitivity of all observables with the exception of $ΔM_K$ to the CKM parameter $|V_{cb}|$ and list a number of $|V_{cb}|$-independent ratios within the SM which exhibit rather different dependences on the angles $β$ and $γ$ of the unitarity triangle. The particular role of these decays in probing very short distance scales far beyond the ones explored at the LHC is emphasized. In this context the role of the Standard Model Effective Field Theory (SMEFT) is very important. We also address briefly the issue of the footprints of Majorana neutrinos in $K^+\rightarrowπ^+ν\barν$ and $K_L\rightarrowπ^0ν\barν$.