The existence of an axion is doubly motivated: it solves a fundamental problem of the Standard Model of elementary particles and it is a leading dark matter candidate. The axion was postulated in 1977, shortly after the Standard Model became widely accepted, to explain why the strong interactions are parity invariant. In 1983, it was shown that axions are abundantly produced in the early universe and would appear as dark matter today. Thus, the existence of an axion has interesting implications for cosmology, astrophysics and particle physics. Searches for other related hypothetical elementary particles such as axion-like particles, paraphotons and chameleons, are also reviewed.