  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 of
the parameters of chiral perturbation theory. The dispersion relation for
$K_{\ell4}$ provides a resummation of $\pi\pi$- and $K\pi$-rescattering
effects. In contrast to a pure chiral treatment, it reproduces the observed
curvature of one of the form factors. The matching of the dispersion relation
to the chiral representation of the form factors allows the extraction of the
values of three low-energy constants.
  Hadronic light-by-light scattering appears as a virtual process in the
calculation of the anomalous magnetic moment of the muon $(g-2)_\mu$. For more
than a decade, a discrepancy of about $3\sigma$ has persisted between the
experimental determination and the standard-model prediction of the
$(g-2)_\mu$. It is expected that within a few years hadronic light-by-light
scattering will dominate the uncertainty of the theory prediction of the
$(g-2)_\mu$. So far, only model calculations of the hadronic light-by-light
contribution are available. However, in view of forthcoming $(g-2)_\mu$
experiments at Fermilab and J-PARC it is crucial that the hadronic
light-by-light calculation can be improved systematically. The dispersive
description presented here provides a formalism for a data-driven determination
of hadronic light-by-light scattering and hence opens up an avenue towards a
model-independent evaluation of the $(g-2)_\mu$.
