 In 1966, when he was thirty-seven years old, he was awarded the Fields Medal, for his work in developing K-theory, a generalized Lefschetz fixed-point theorem and the Atiyah–Singer theorem, for which he also won the Abel Prize jointly with Isadore Singer in 2004.Among other prizes he has received are the Royal Medal of the Royal Society in 1968, the De Morgan Medal of the London Mathematical Society in 1980, the Antonio Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Copley Medal of the Royal Society in 1988, the Benjamin Franklin Medal for Distinguished Achievement in the Sciences of the American Philosophical Society in 1993, the Jawaharlal Nehru Birth Centenary Medal
of the Indian National Science Academy in 1993, the President's Medal from the Institute of Physics in 2008, the Grande Médaille of the French Academy of Sciences in 2010 and the Grand Officier of the French Légion d'honneur in 2011.He was elected a foreign member of the National Academy of Sciences, the American Academy of Arts and Sciences (1969), the Académie des Sciences, the Akademie Leopoldina, the Royal Swedish Academy, the Royal Irish Academy, the Royal Society of Edinburgh, the American Philosophical Society, the Indian National Science Academy, the Chinese Academy of Science, the Australian Academy of Science, the Russian Academy of Science, the Ukrainian Academy of Science, the Georgian Academy of Science, the Venezuela Academy of Science, the Norwegian Academy of Science and Letters, the Royal Spanish Academy of Science, the Accademia dei Lincei and the Moscow Mathematical Society.