Clayton was elected to Phi Beta Kappa during his third year as a student at Southern Methodist University. He was awarded many supporting fellowships: National Science Foundation Predoctoral Fellow (1956–58); Alfred P. Sloan Foundation Fellow (1966–68); Fulbright Fellow (1979–80); Fellow of St. Mary's College, Durham University (1987); SERC Senior Visiting Fellow, The Open University, Milton Keynes, U.K. (1993). In 1993 Clayton was named Distinguished Alumnus of Southern Methodist University, thirty-seven years after his BS degree there.
