  An effective field theory is used to give a model-independent description of
Compton scattering at energies comparable to the pion mass. The amplitudes for
scattering on the proton and the deuteron, calculated to fourth order in small
momenta in chiral perturbation theory, contain four undetermined parameters
that are in one-to-one correspondence with the nucleon polarizabilities. These
polarizabilities are extracted from fits to data on elastic photon scattering
on hydrogen and deuterium. For the proton we find: alpha_p = 12.1 +/- 1.1 +/-
0.5, beta_p = 3.4 +/- 1.1 +/- 0.1, both in units of 10^{-4} fm^3. For the
isoscalar polarizabilities we obtain: alpha_N = 13.0 +/- 1.9 +3.9/-1.5 (in the
same units) while beta_N is consistent with zero, within sizeable error bars.
