A major goal of axion searches is to reach inside the parameter space region of realistic axion models. Currently, the boundaries of this region depend on somewhat arbitrary criteria, and it would be desirable to specify them in terms of precise phenomenological requirements. I will start with a brief description of the strong CP problem, of the Peccei-Quinn solution predicting the existence of the axion, and of axions as cold Dark Matter. I will then consider the hadronic axion models which involve new heavy quarks Q. By requiring that i) the Q are sufficiently short lived to avoid issues with long lived strongly interacting relics, ii) no Landau poles are induced below the Planck scale, fifteen cases are selected, which define a phenomenologically preferred axion window.