  We calculate the decay distributions for inclusive B -> X_s gamma and B ->
X_u l nu decays in the endpoint region, where radiative corrections are large.
The computation is done using effective field theory methods. The matching
coefficients are computed to O(alpha_s), and the anomalous dimensions to
next-to-leading order. The final expressions for the differential decay spectra
include the complete O(alpha_s) corrections, and sum the leading and
next-to-leading Sudakov series. We present results for regions of phase space
where the shape function can be expanded in local operators, and give the
matching coefficients of the resulting enhanced non-perturbative effects to
order alpha_s. We show that moments of the shape function {\it are not} given
by moments of local operators once perturbative effects are included, explain
why the shape function and its moments satisfy different renormalization group
equations, and contrast this with the situation for deep inelastic scattering.
We show that there are large perturbative corrections in the usual definition
of the shape function. This renders incorrect previous prescriptions for
combining radiative corrections with the shape function.
