  Within a systematic approach based on nonrelativistic quantum electrodynamics
(NRQED), we derive the one-loop self-energy correction of order alpha
(Zalpha)^4 to the bound-electron g factor. In combination with numerical data,
this analytic result improves theoretical predictions for the self-energy
correction for carbon and oxygen by an order of magnitude. Basing on one-loop
calculations, we obtain the logarithmic two-loop contribution of order alpha^2
(Zalpha)^4 ln[(Zalpha)^-2] and the dominant part of the corresponding constant
term. The results obtained improve the accuracy of the theoretical predictions
for the 1S bound-electron g factor and influence the value of the electron mass
determined from g factor measurements.
