  The Standard Model (SM) of particle physics is a big success. However, it
lacks explanations for cosmic inflation, the matter-anti-matter asymmetry of
the Universe, dark matter, neutrino oscillations, and the feebleness of CP
violation in the strong interactions. The latter may be explained by a complex
scalar field charged under a spontaneously broken global U(1) Peccei-Quinn (PQ)
symmetry. Moreover, the pseudo Nambu-Goldstone boson of this breaking -- the
axion -- may play the role of the dark matter. Furthermore, the modulus of the
PQ field is a candidate for driving inflation. If additionally three extra SM
singlet neutrinos (whose mass is induced by the PQ field) are included, the
five aforementioned problems can be addressed at once. We review the SM
extension dubbed SMASH -for SM-Axion-Seesaw-Higgs portal inflation-, discuss
its predictions and tests in astrophysics, cosmology, and laboratory
experiments. Variants of SMASH are also considered and commented on.
