Shaw studied at the University of Manchester where he gained a BSc and a PhD.  After three years in the Civil Service at the Torry Research Station in Aberdeen and five years at ICI he was appointed Lecturer in the Department of Inorganic and Structural Chemistry at the University of Leeds in 1962.  He was promoted to Reader in 1966 and to a Personal Chair in 1971.  He was made head of the university's department of Inorganic Chemistry in 1990 and Emeritus Professor upon his retirement in 1995.
Shaw was at various times a Visiting Professor at the University of Western Ontario, Carnegie Mellon University (1969), the American National University (1983) and the University of Auckland (1986).  He was Lecturer at the Royal Society of Chemistry in 1987–88.