  For several decades, the $\sigma/f_0(980)$ and $\kappa/K_0^*(700)$ resonances
have been subject to long-standing debate. Both their existence and properties
were controversial until very recently. In this tutorial review we compare
model-independent dispersive and analytic techniques versus unitarized Chiral
Perturbation Theory, when applied to the lightest scalar mesons
$\sigma/f_0(980)$ and $\kappa/K_0^*(700)$. Generically, the former have settled
the long-standing controversy about the existence of these states, providing a
precise determination of their parameters, whereas unitarization of chiral
effective theories allows us to understand their nature, spectroscopic
classification and dependence on QCD parameters. Here we review in a
pedagogical way their uses, advantages and caveats.
