  We discuss the application of an effective field theory (EFT) which
incorporates the chiral symmetry of QCD to Compton scattering from the proton
and deuteron. We describe the chiral EFT analysis of the proton Compton
scattering database presented in our recent review (arXiv:1203.6834), which
gives: alpha^{(p)}=10.5 +/- 0.5(stat) +/- 0.8(theory); beta^{(p)}= 2.7 +/-
0.5(stat) +/- 0.8(theory), for the electric and magnetic dipole polarizability
of the proton. We also summarize the chiral EFT analysis of the world data on
coherent Compton scattering from deuterium presented in arXiv:1203.6834. That
yields: alpha^{(s)}=10.5 +/- 2.0(stat) +/- 0.8(theory); beta^{(s)}=3.6 +/-
1.0(stat) +/- 0.8(theory).
