CP, or not CP, that is the question...

Abstract: Motivated by recent claims questioning the existence of strong CP violation, we present a pedagogical review of CP violation in Quantum Chromodynamics (QCD). Using fundamental properties of the QCD partition function, we analyze the dependence of the chiral quark and CP violating gluon condensates on the theta parameter and the quark masses in the chiral limit. We show explicitly how CP violation arises, clarify the role of the axial U(1) anomaly and the ordering of the infinite-volume limit, and discuss the conditions under which CP symmetry may or may not be realized, including in the large-N framework. Our results reaffirm the presence of strong CP violation for physically relevant parameters and thus the theoretical basis of the strong CP problem and axion physics.