  We consider the effects of inverse Compton scattering of synchrotron photons
from relativistic electrons in GRB afterglows. We compute the spectrum of the
inverse Compton emission and find that it can dominate the total cooling rate
of the afterglow for several months or even years after the initial explosion.
We demonstrate that the presence of strong inverse Compton cooling can be
deduced from the effect it has on the time-evolution of the cooling break in
the synchrotron spectral component, and therefore on the optical and X-ray
afterglow lightcurves. We then show how the physical interpretation of the
observed characteristics of the synchrotron spectrum must be modified to take
into consideration this extra source of cooling, and give a revised
prescription for computing physical parameters characterizing the expanding
shock wave from the observed quantities. We find that for a given set of
observables (synchrotron break frequencies and fluxes) there is either no
consistent physical interpretation or two of them. Finally we discuss the
prospects of directly detecting the inverse Compton emission with Chandra. We
argue that such a detection is possible for GRBs exploding in a reasonably
dense (n>1 cm^-3) medium.
