  We review the cross sections for low-energy $\pi^- \gamma$ reactions in the
framework of chiral perturbation theory. Charged pion Compton scattering,
$\pi^- \gamma\to \pi^-\gamma$, is considered up to one-loop order where the
pion's internal structure enters through the difference of the electric and
magnetic pion polarizability, $\alpha_\pi - \beta_\pi$. The ongoing COMPASS
experiment aims at measuring this important structure constant with high
statistics using the Primakoff effect. In the same way, the reaction $\pi^-
\gamma \to \pi^- \pi^0$ serves as a test of the QCD chiral anomaly (via the
$\gamma 3\pi$-coupling constant $F_{\gamma3\pi}$). Furthermore, we calculate
the total cross sections for neutral and charged pion-pair production, $\pi^-
\gamma \to \pi^- \pi^0\pi^0$ and $\pi^- \gamma \to \pi^-\pi^+\pi^-$, which are
governed by the chiral $\pi\pi$-interaction. Finally, we investigate the
radiative (correction) process $\pi^- \gamma \to \pi^- \gamma \gamma$ and
calculate the corresponding two-photon mass spectrum. This information will be
useful for analyzing the $\pi^-$ Primakoff scattering events with photons in
the final state.
