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Adams was born at Lidcot farm, Cornwall, in 1819, the eldest son of a tenant farmer. He developed an early interest in astronomy and in 1831 was sent to his cousin's academy, where he distinguished himself in classics and spent his spare time on astronomy and mathematics. Adams's progress was such that his parents decided that he should be sent to university, and in October 1839 he sat for examinations at St John's College and won a sizarship. In July 1841, at the end of his second year, he wrote himself the following memorandum: 'Formed a design ... of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus, which are yet unaccounted for; in order to find whether they may be attributed to the action of an undiscovered planet beyond it ...' Having won the highest mathematical prizes in his college, Adams graduated in 1843 as senior wrangler and won a fellowship. He could now return to his deferred investigation of Uranus. By October 1843 Adams, aged just 24, had arrived at a solution of the inverse perturbation problem and although his first result was approximate, it convinced him that the disturbances of Uranus were due to an undiscovered planet.

In February 1844 Adams applied to the astronomer royal, Sir George Biddell Airy, for more exact data on Uranus. With Airy's figures Adams then computed values for the elliptic elements, mass, and heliocentric longitude of the hypothetical planet. He gave his results to James Challis, Director of the Cambridge Observatory, in September 1845, and after two unsuccessful attempts to present his work to Airy in person, left a copy at the Royal Observatory in October. Airy replied to Adams a few weeks later but did not institute a search for the planet until July 1846.

In the meantime the French astronomer Urbain Jean Joseph Le Verrier had independently published several papers on Uranus and reached the same conclusions as Adams regarding an exterior planet. It was as a result of Le Verrier's efforts that Johann Gottfried Galle, of the Berlin Observatory, discovered Neptune on 23 September 1846, less than one degree distant from where Le Verrier had predicted it would lie. While Le Verrier was showered with honours, Adams's earlier prediction, which agreed closely with Le Verrier's, remained unpublished. First publicised in a letter from Sir John Herschel to the London Athenaeum in October 1846 it provoked a long and bitter controversy over priority of discovery and the issue became a public sensation. Adams and Le Verrier themselves, however, met at Oxford in 1847 and became good friends. Adams was offered a knighthood by Queen Victoria in 1847 but declined. In 1848 the Adams Prize was founded at Cambridge and the Royal Society awarded him its highest award, the Copley Medal.

Adams was elected President of the Royal Astronomical Society in 1851 and began work on lunar theory. On the expiration of his fellowship at St John's he moved to Pembroke College in 1853 and shortly afterwards presented to the Royal Society a remarkable paper on the secular acceleration of the Moon's mean motion, showing Laplace's 1788 solution to be incorrect. While this provoked a sharp scientific controversy, Adams was later proved to be right.

In 1858 Adams became Professor of Mathematics at St Andrew's University but returned to Cambridge in 1859 to become Lowndean Professor of Astronomy and Geometry. In 1861 he took over as Director of the Cambridge Observatory and two years later married Eliza Bruce of Dublin. In the 1860s and 70s he undertook work on the Leonid system, observations for the Astronomische Gesellschaft program, work on Bernoulli numbers and Euler's constant, and the arrangement and cataloguing of Newton's mathematical papers, presented to Cambridge University by Lord Portsmouth. While much of Adams's later work has been superseded, as the co-discoverer of Neptune he occupies a special place in the history of science.