  The decays $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$, being the
theoretically cleanest rare decays of mesons, are very sensitive probes of New
Physics. In view of the excellent prospects of reaching the Standard Model
sensitivity for $K^+\to\pi^+\nu\bar\nu$ by the NA62 experiment at CERN and for
$K_L\to\pi^0\nu\bar\nu$ by the KOTO experiment at J-PARC, we study them in the
simplest extensions of the SM in which stringent correlations between these two
decays and other flavour observables are present. We first consider simple
models with tree-level Z and Z' contributions in which either MFV or a $U(2)^3$
symmetry is imposed on the quark flavour-violating couplings. We then compare
the resulting correlations with those present in generic models in which the
latter couplings are arbitrary, subject to constraints from $\Delta F=2$
processes, electroweak and collider data. Of particular interest are the
correlations with $\epsilon'/\epsilon$ and $K_L\to\mu^+\mu^-$ which limit the
size of NP contributions to $K^+\to\pi^+\nu\bar\nu$ and
$K_L\to\pi^0\nu\bar\nu$, depending on the Dirac structure of couplings and the
relevant operators. But in MFV also the constraint from $B_s\to\mu^+\mu^-$
turns out to be important. We take into account the recent results from lattice
QCD and large N approach that indicate $\epsilon'/\epsilon$ in the SM to be
significantly below the data. While in many models the enhancement of
$\epsilon'/\epsilon$ implies a suppression of $K_L\to\pi^0\nu\bar\nu$, we
present two models in which these observables can be simultaneously enhanced
relative to SM predictions. A correlation between $K^+\to\pi^+\nu\bar\nu$ and
$B\to K(K^*)\mu^+\mu^-$, found by us in the simple models considered here,
should be of interest for NA62 and LHCb experimentalists at CERN in the coming
years. The one with $B\to K(K^*)\nu\bar\nu$ will be tested at Belle II.
