Axions, strong and weak CP, and KNP inflation

I review the ideas leading to the QCD axion and also comment on the Jarlskog determinant describing the observed weak CP violation, and the axion-related Kim-Nilles-Peloso inflation. All of these use pseudoscalars, and the underlying principle is the discrete gauge symmetry either in the bottom-up or top-down approaches. Here, the effects of gravity are required to be unimportant in the low energy effective theory. String compactification is safe from the gravity spoil of global symmetries and some examples from string compactification are commented.