  Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup
defined over an algebraically closed field of characteristic $p>0$. In this
paper, the authors study the existence of generic $G$-cohomology and its
stability with rational $G$-cohomology groups via the use of methods from the
authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology
groups are presented. Furthermore, vanishing ranges for the associated finite
group cohomology of $G({\mathbb F}_{q})$ are established which generalizes
earlier work of Hiller, in addition to stability ranges for generic cohomology
which improves on seminal work of Cline, Parshall, Scott and van der Kallen.
