© 2019 IOP Publishing Ltd and Sissa Medialab. We review the physics potential of a next generation search for solar axions: the International Axion Observatory (IAXO) . Endowed with a sensitivity to discover axion-like particles (ALPs) with a coupling to photons as small as gaγ∼ 10-12 GeV-1, or to electrons gae∼10-13, IAXO has the potential to find the QCD axion in the 1 meV∼1 eV mass range where it solves the strong CP problem, can account for the cold dark matter of the Universe and be responsible for the anomalous cooling observed in a number of stellar systems. At the same time, IAXO will have enough sensitivity to detect lower mass axions invoked to explain: 1) the origin of the anomalous "transparency" of the Universe to gamma-rays, 2) the observed soft X-ray excess from galaxy clusters or 3) some inflationary models. In addition, we review string theory axions with parameters accessible by IAXO and discuss their potential role in cosmology as Dark Matter and Dark Radiation as well as their connections to the above mentioned conundrums.