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We analyse the $\theta$-angle physics associated to extensions of thestandard model of particle interactions featuring new strongly coupled sectors.We start by providing a pedagogical review of the $\theta$-angle physics forQuantum Chromodynamics (QCD) including also the axion properties. We then moveto analyse composite extensions of the standard model elucidating the interplaybetween the new $\theta$-angle with the QCD one. We consider first QCD-likedynamics and then generalise it to consider several kinds of new stronglycoupled gauge theories with fermions transforming according to different matterrepresentations. Our analysis is of immediate use for different models ofcomposite Higgs dynamics, composite dark matter and inflation.

The Physics of the $\theta$-angle for Composite Extensions of the Standard Model