In the primordial Universe the axion particles, which solve in an elegant way the CP problem in QCD, can be produced both thermally, contributing to the hot dark matter of the Universe, or not thermally, contributing to the cold dark matter. The thermal axions have masses strongly de- generate with those of the three active neutrinos, as they leave identical signatures in the different cosmological observables. In addition, while still relativistic states, they also contribute to the rel- ativistic degrees of freedom, contributing to the dark radiation. Here, we will focus on the recent constraints from cosmology for the thermal axion mass, in the standard cosmological scenario ΛCDM as well as his extensions, but also taking into account the possibility that the primordial power spectrum could assume a more general shape than the usual power law description.