  I present analytical expressions for the collinear and soft gluon corrections
to Higgs production via the process b bbar -> H as well as gg -> H through
next-to-next-to-next-to-leading order (NNNLO). The soft corrections are
complete while the collinear corrections include leading and some subleading
logarithms. Numerical results at the Tevatron and the LHC are presented,
primarily for b bbar -> H. It is shown that the collinear terms greatly improve
the soft and virtual approximation at next-to-leading order (NLO) and
next-to-next-to-leading order (NNLO), especially when subleading terms are
included. The NNNLO collinear and soft corrections provide significant
enhancements to the total cross section. I also provide expressions for the
collinear and soft corrections through NNNLO for the related Drell-Yan process.
