  I discuss how anomalies affect classical symmetries and how, in turn, the
non-trivial nature of the gauge theory vacuum makes these quantum corrections
troublesome. Although no solution seems in sight for the cosmological constant
problem, I examine three possible approaches to the strong CP problem involving
vacuum dynamics, an additional chiral symmetry, and the possibility of
spontaneous CP or P breaking. All of these "solutions" have their own problems
and suggest that, at a deep level, we do not understand the nature of CP
violation. Nevertheless, it remains extremely important to search for
experimental signals predicted by these theoretical "solutions", like invisible
axions.
