Anthony Kelly, known as Tony, was born in Hillingdon, Middlesex on 25 January 1929. His mother Violet Vaughan was a nurse who converted to Catholicism on marriage and his father, Group Captain Vincent Gerald Kelly, was a veteran of the First World War trenches who had witnessed his brother’s death at the Somme. Kelly obtained two first class honours degrees in General and Special (Physics) from Reading University and a Ph.D. at Trinity College Cambridge in 1953. His thesis was concerned with the plastic deformation of metals examined by X-ray microbeam diffraction.

Kelly then worked at the University of Illinois (1953-1955) and at Birmingham University (1955) before spending three years as an Assistant then Associate Professor of Metallurgy and Materials Science at Northwestern University, Chicago (1956-1959). He returned to Cambridge in 1959 as a Founding Fellow of Churchill College and as a University Lecturer where, in addition to teaching, he built up a research group working on the strength of materials.

From 1967-1975, Kelly took on a number of managerial roles within government science, becoming Superintendent of the Materials Division of the National Physical Laboratory (NPL) and then Deputy Director in 1969. At the NPL, Kelly developed his knowledge of the strength of materials and his research focused on high performing composite materials (such as plastic reinforced with carbon fibre) which are used in vehicles on land, sea and air, in buildings, bridges and engines. Kelly received an Sc.D. from the University of Cambridge in 1968 and was elected a Fellow of the Royal Society in 1973 at the age of 43.

In 1975, Kelly became the second Vice-Chancellor of the University of Surrey, a position he held until 1994. Student numbers increased dramatically during this period from 3,000 to 7,000 and annual income also rose from £6 to £63 million. He responded dramatically to serious financial pressure resulting from the University Grants Committee cuts of 1981 by closing a number of departments and increasing income from non-government sources.

Kelly continued to make significant contributions to materials science during and after his time as Vice-Chancellor of Surrey University. He is sometimes known as "the father of composite materials" and recognized for having founded and developed the field of strong solids made from composite materials, now extensively used in industry.

In 1994 Kelly retired and returned to Cambridge as Emeritus Professor and distinguished research fellow in the department of materials science and technology. He was also closely involved with Churchill College and edited the "Churchill Review" from 1997-2010. In 2010, Kelly led a successful effort to get 43 Fellows of the Royal Society to petition the Council to modify its public stance on climate science and became a member of the Academic Advisory Council to the Global Warming Policy Foundation.

Kelly married Christina (d.1997) with whom he had 3 sons and 1 daughter. He died on 3 June 2014.