Measuring the polarizabilities of the proton and pion with photon and hadron beams

Polarizabilities provide an important test point for models of hadron structure, as well as potentially helping to resolve two of the most outstanding anomalies in nuclear physics, the muon g-2 and the proton charge radius puzzles. Recent progress in measurements of the proton scalar polarizabilities, $\alpha$ and $\beta$, and spin-polarizabilities from polarized Compton scattering experiments at Mainz and TUNL/HIGS are presented. This new generation of Compton scattering experiments utilize linear/circularly polarized incident photons and polarized targets. For many years the charged pion polarizability ranked among the most important tests of ChPT unresolved by experiment. A new result for the charged pion polarizability from the Compass experiment is presented, and the outlook for a precision measurement of the charged pion polarizability at JLab through the $\gamma \gamma \rightarrow \pi^+ \pi^-$ reaction is discussed.