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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. Theunprecedented statistics of present and future Hadron Experiments, modernLattice QCD calculations, and the wealth of new states and decays, require suchprecise and model-independent analyses to describe final state interactions. Wereview the existing and often conflicting data and explain in detail thederivation of the relevant dispersion relations, maximizing their applicabilityrange. Next, we review and extend the caveats on some data, showing theirinconsistency with dispersion relations. Our main result is the derivation andcompilation of precise amplitude parameterizations constrained by several $\piK\to \pi K$ and $\pi \pi \to K \bar{K}$ dispersion relations. These constrainedparameterizations are easily implementable and provide the rigor and accuracyneeded for modern experimental and phenomenological Hadron Physics. Asapplications, after reviewing their status and interest, we will provide newprecise threshold and subthreshold parameters and review our dispersivedetermination of the controversial $\kappa/K^*_0(700)$ resonance and otherlight-strange resonances.

physics reports

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 $K^*_0(700)$