  In this paper, we prove some a priori stability estimates (in weighted
Sobolev spaces) for the spatially homogeneous Boltzmann equation without
angular cutoff (covering every physical collision kernels). These estimates are
conditioned to some regularity estimates on the solutions, and therefore reduce
the stability and uniqueness issue to the one of proving suitable regularity
bounds on the solutions. We then prove such regularity bounds for a class of
interactions including the so-called (non cutoff and non mollified) hard
potentials and moderately soft potentials. In particular, we obtain the first
result of global existence and uniqueness for these long-range interactions.
