In view of future plans for accurate measurements of the theoretically clean branching ratios Br(K+ -> pi+ nu nu) and Br(KL -> pi0 nu nu), that should take place in the next decade, we collect the relevant formulae for quantities of interest and analyze their theoretical and parametric uncertainties. We point out that in addition to the angle beta in the unitarity triangle (UT) also the angle gamma can in principle be determined from these decays with respectable precision and emphasize in this context the importance of the recent NNLO QCD calculation of the charm contribution to K+ -> pi+ nu nu and of the improved estimate of the long distance contribution by means of chiral perturbation theory. In addition to known expressions we present several new ones that should allow transparent tests of the Standard Model (SM) and of its extensions. While our presentation is centered around the SM, we also discuss models with minimal flavour violation and scenarios with new complex phases in decay amplitudes and meson mixing. We give a brief review of existing results within specific extensions of the SM, in particular the Littlest Higgs Model with T-parity, Z' models, the MSSM and a model with one universal extra dimension. We derive a new "golden" relation between B and K systems that involves (beta,gamma) and Br(KL -> pi0 nu nu) and investigate the virtues of (R_t,beta), (R_b,gamma), (beta,gamma) and (etabar,gamma) strategies for the UT in the context of K -> pi nu nu decays with the goal of testing the SM and its extensions.

Comment: 56 pages, 18 figures, Section on Long Distance Contributions, 2 Figures and few References added, Uses Rev Mod Phys Style; Includes new results of NNLO calculation as well as matrix elements, extended and modified sections on new physics