The Return of Kaon Flavour Physics

Kaon flavour physics has played in the 1960s and 1970s a very important role in the construction of the Standard Model (SM) and in the 1980s and 1990s in SM tests with the help of CP violation in $K_L\to\pi\pi$ decays represented by $\varepsilon_K$ and the ratio $\varepsilon'/\varepsilon$. In this millennium this role has been taken over by $B_{s,d}$ and $D$ mesons. However there is no doubt that in the coming years we will witness the return of kaon flavour physics with the highlights being the measurements of the theoretically clean branching ratios for the rare decays $K^+\rightarrow \pi^+\nu\bar\nu$ and $K_{L}\rightarrow\pi^0\nu\bar\nu$ and the improved SM predictions for the ratio $\varepsilon'/\varepsilon$, for $\varepsilon_K$ and the $K^0-\bar K^0$ mixing mass difference $\Delta M_K$. Theoretical progress on the decays $K_{L,S}\to\mu^+\mu^-$ and $K_L\to\pi^0\ell^+\ell^-$ is also expected. They all are very sensitive to new physics (NP) contributions and the correlations between them should help us to identify new dynamics at very short distance scales. These studies will be enriched when theory on the $K\to\pi\pi$ isospin amplitudes ${\rm Re} A_0$ and ${\rm Re} A_2$ improves. This talk summarizes several aspects of this exciting field. In particular we emphasize the role of the Dual QCD approach in getting the insight into the numerical Lattice QCD results on $K^0-\bar K^0$ mixing and $K\to\pi\pi$ decays.