  We study bounds on nilpotence in H*(BG), the mod p cohomology of the
classifying space of a compact Lie group G. Part of this is a report of our
previous work on this problem, updated to reflect the consequences of Peter
Symonds recent verification of Dave Benson's Regularity Conjecture. New results
are given for finite p--groups, leading to good bounds on nilpotence in H*(BP)
determined by the subgroup structure of the p--group P.
