  We study Compton scattering in systems with A=1 and 2 using chiral
perturbation theory up to fourth order. For the proton we fit the two
undetermined parameters in the O(Q^4) $\gamma$p amplitude of McGovern to
experimental data in the region $\omega,\sqrt{|t|} \leq 180$ MeV, obtaining a
chi^2/d.o.f. of 133/113. This yields a model-independent extraction of proton
polarizabilities based solely on low-energy data: alpha_p=12.1 +/- 1.1 (stat.)
+/- 0.5 (theory) and beta_p=3.4 +/- 1.1 (stat.) +/- 0.1 (theory), both in units
of 10^{-4} fm^3. We also compute Compton scattering on deuterium to O(Q^4). The
$\gamma$d amplitude is a sum of one- and two-nucleon mechanisms, and contains
two undetermined parameters, which are related to the isoscalar nucleon
polarizabilities. We fit data points from three recent $\gamma$d scattering
experiments with a chi^2/d.o.f.=26.6/20, and find alpha_N=13.0 +/- 1.9 (stat.)
+3.9/-1.5 (theory) and a beta_N that is consistent with zero within sizeable
error bars.
