In view of future plans for accurate measurements of the theoretically clean branching ratios $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ and $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$, which should occur in the next decade, the relevant formulas for quantities of interest are collected and their theoretical and parametric uncertainties are analyzed. In addition to the angle $\beta$ in the unitarity triangle (UT), the angle $\gamma$ can also be determined from these decays with respectable precision and in this context the importance of the recent NNLO QCD calculation on the charm contribution to $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and of the improved estimate on the long-distance contribution by means of chiral perturbation theory are presented. In addition to known expressions, several new ones that should allow transparent tests of the standard model (SM) and of its extensions are presented. While the review is centered around the SM, models with minimal flavor violation and scenarios with new complex phases in decay amplitudes and meson mixing are also discussed. A review of existing results within specific extensions of the SM, in particular the littlest Higgs model with $T$-parity, $Z^{\prime }$ models, the MSSM, and a model with one universal extra dimension are given. A new “golden” relation between $B$ and $K$ systems is derived that involves $(\beta ,\gamma )$ and $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$, and the virtues of $(R_t,\beta )$, $(R_b,\gamma )$, $(\beta ,\gamma )$, and $(\overline{\eta },\gamma )$ strategies for the UT in the context of $K\to \pi \nu \overline{\nu }$ decays with the goal of testing the SM and its extensions are investigated.

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