The absence of strong CP violation is one of the most puzzling experimental facts of modern particle physics. In the Standard Model (SM), such a violation appears naturally, and should even be relatively large, but this is in contradiction with the current very tight limits on the neutron electric dipole moment. No solution to this puzzle appears possible within the SM, and some new physics is unavoidable. The simplest models designed to tackle specifically the strong CP puzzle are based on axions, and are the subject of this talk.
Specifically, after a brief overview of the strong CP puzzle, we will discuss triangle anomalies which are an essential ingredient of axion physics. Then, we will turn to axion models, and in particular to their predictions for the axion to SM gauge boson couplings. Our goal will be to prove that these couplings always arise from non-anomalous fermion loops, and are thus independent of the anomaly structure of the model. The reason for this mismatch is traced back to triangle Feynman diagrams sensitive to the anomalous breaking of the vector Ward identity, and is ultimately related to the conservation of baryon and lepton numbers. Though we will discuss in details only true axion models, we will end by commenting on the implications for axion-like particle searches.