We construct the Roy-Steiner equations for pion Compton scattering. The Roy-Steiner equations
respect analyticity and unitarity requirements, gauge invariance, as well as crossing symmetry.
To suppress the dependence on the high-energy region, we consider once- and twice-subtracted
versions of the equations, where the subtraction constants are identified with dipole and
quadrupole pion polarizabilities. As an application we study the gamma gamma--> pi pi partial waves
by a Muskhelishvili-Omn'es representation with finite matching point, and discuss the consequences
for the two-photon coupling of the sigma resonance and its relation to pion polarizabilities.