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In this thesis, I present dispersive treatments of two hadronic processes:the semileptonic kaon decay $K_{\ell4}$ and hadronic light-by-light scattering.  The $K_{\ell4}$ decay is one of the best sources of information on some ofthe parameters of chiral perturbation theory. The dispersion relation for$K_{\ell4}$ provides a resummation of $\pi\pi$- and $K\pi$-rescatteringeffects. In contrast to a pure chiral treatment, it reproduces the observedcurvature of one of the form factors. The matching of the dispersion relationto the chiral representation of the form factors allows the extraction of thevalues of three low-energy constants.  Hadronic light-by-light scattering appears as a virtual process in thecalculation of the anomalous magnetic moment of the muon $(g-2)_\mu$. For morethan a decade, a discrepancy of about $3\sigma$ has persisted between theexperimental determination and the standard-model prediction of the$(g-2)_\mu$. It is expected that within a few years hadronic light-by-lightscattering will dominate the uncertainty of the theory prediction of the$(g-2)_\mu$. So far, only model calculations of the hadronic light-by-lightcontribution are available. However, in view of forthcoming $(g-2)_\mu$experiments at Fermilab and J-PARC it is crucial that the hadroniclight-by-light calculation can be improved systematically. The dispersivedescription presented here provides a formalism for a data-driven determinationof hadronic light-by-light scattering and hence opens up an avenue towards amodel-independent evaluation of the $(g-2)_\mu$.

Dispersive Treatments of $K_{\ell4}$ Decays and Hadronic Light-by-Light Scattering