The processes involved were elucidated over a number of years from early in the 20th century, when it was first realised that the energy released from nuclear reactions accounted for the longevity of the Sun as a source of heat and light.
In 1920, Arthur Eddington, on the basis of the precise mesurements of atoms by F.W Aston, was the first to suggest that stars obtained their energy from nuclear fusion of hydrogen to helium. In 1928, George Gamow derived what is now called the Gamow factor, a quantum-mechanical formula that gave the probability of bringing two nuclei sufficiently close for the strong nuclear force to overcome the Coulomb barrier. The Gamow factor was used in the decade that followed by Atkinson and Houtermans and later by Gamow himself and Teller to derive the rate at which nuclear reactions would proceed at the high temperatures believed to exist in stellar interiors.
In 1939, in a paper entitled "Energy Production in Stars", Hans Bethe analyzed the different possibilities for reactions by which hydrogen is fused into helium. He selected two process that he believed are the source of energy in stars. The first one, the proton-proton chain, is the dominant energy source in stars low masses about like the Sun or smaller. The second process, the carbon-nitrogen-oxygen cycle, which was also considered by Carl von Weizsäcker in 1938, is most important in more massive stars.
Later, many important details were added to Bethe's theory, like the publication of a celebrated paper in 1957 by Burbidge, Burbidge, Fowler and Hoyle. This latter work collected and refined earlier researches into a coherent picture that accounted for the observed relative abundances of the elements.
The most important such reactions are: