  We prove the existence and exponential decay of global in time strong
solutions to the Boltzmann equation without any angular cut-off, i.e., for
long-range interactions. We consider perturbations of the Maxwellian
equilibrium states and include the physical cross-sections arising from an
inverse-power intermolecular potential $r^{-(p-1)}$ with $p>3$, and more
generally, the full range of angular singularities $s=\nu/2 \in(0,1)$. These
appear to be the first unique global solutions to this fundamentally important
model, which grants a basic example where a range of geometric fractional
derivatives occur in a physical model of the natural world. Our methods provide
a new understanding of the effects of grazing collisions in the Boltzmann
theory.
