Dispersive $\pi K\to \pi K$ and $\pi \pi \to K \bar{K}$ amplitudes from scattering data, threshold parameters and the lightest strange resonance $\kappa$ or $K^*_0(700)$

We discuss the simultaneous dispersive analyses of $\pi K\to \pi K$ and $\pi \pi \to K \bar{K}$ scattering data and the $\kappa/K^*_0(700)$ resonance. The unprecedented statistics of present and future Hadron Experiments, modern Lattice QCD calculations, and the wealth of new states and decays, require such precise and model-independent analyses to describe final state interactions. We review the existing and often conflicting data and explain in detail the derivation of the relevant dispersion relations, maximizing their applicability range. Next, we review and extend the caveats on some data, showing their inconsistency with dispersion relations. Our main result is the derivation and compilation of precise amplitude parameterizations constrained by several $\pi K\to \pi K$ and $\pi \pi \to K \bar{K}$ dispersion relations. These constrained parameterizations are easily implementable and provide the rigor and accuracy needed for modern experimental and phenomenological Hadron Physics. As applications, after reviewing their status and interest, we will provide new precise threshold and subthreshold parameters and review our dispersive determination of the controversial $\kappa/K^*_0(700)$ resonance and other light-strange resonances.