  We introduce a fast Fourier spectral method for the spatially homogeneous
Boltzmann equation with non-cutoff collision kernels. Such kernels contain
non-integrable singularity in the deviation angle which arise in a wide range
of interaction potentials (e.g., the inverse power law potentials). Albeit more
physical, the non-cutoff kernels bring a lot of difficulties in both analysis
and numerics, hence are often cut off in most studies (the well-known Grad's
angular cutoff assumption). We demonstrate that the general framework of the
fast Fourier spectral method can be extended to handle the non-cutoff kernels,
achieving the accuracy/efficiency comparable to the cutoff case. We also show
through several numerical examples that the solution to the non-cutoff
Boltzmann equation enjoys the smoothing effect, a striking property absent in
the cutoff case.
