  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.
