He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his Ph.D. from the University of Cambridge in 1956. His thesis, written under the direction of Shaun Wylie, was titled On spectral sequences and self-obstruction invariants. He held the Fielden Chair at the University of Manchester (1964–1970), and became Lowndean Professor of Astronomy and Geometry at the University of Cambridge (1970–1989). He was elected a Fellow of the Royal Society in 1964.
In the 1950s, homotopy theory was at an early stage of development, and unsolved problems abounded. Adams made a number of important theoretical advances in algebraic topology, but his innovations were always motivated by specific problems. Influenced by the French school of Henri Cartan and Jean-Pierre Serre, he reformulated and strengthened their method of killing homotopy groups in spectral sequence terms, creating the basic tool of stable homotopy theory now known as the Adams spectral sequence. This begins with Ext groups calculated over the ring of cohomology operations, which is the Steenrod algebra in the classical case. He used this spectral sequence to attack the celebrated Hopf invariant one problem, which he completely solved in a 1960 paper by making a deep analysis of secondary cohomology operations. The Adams–Novikov spectral sequence is an analogue of the Adams spectral sequence using an extraordinary cohomology theory in place of classical cohomology: it is a computational tool of great potential scope.
Adams was also a pioneer in the application of K-theory. He invented the Adams operations in K-theory, which are derived from the exterior powers; they are now also widely used in purely algebraic contexts. Adams introduced them in a 1962 paper to solve the famous vector fields on spheres problem. Subsequently he used them to investigate the Adams conjecture, which is concerned (in one instance) with the image of the J-homomorphism in the stable homotopy groups of spheres. A later paper of Adams and Michael F. Atiyah uses the Adams operations to give an extremely elegant and much faster version of the above-mentioned Hopf invariant one result.
Adams had many talented students, and was highly influential in the development of algebraic topology in Britain and worldwide. His University of Chicago lectures were published in a 1996 series titled "Chicago Lectures in Mathematics Series", such as Lectures on Exceptional Lie Groups and Stable Homotopy and Generalised Homology ISBN 0-226-00524-0 .