Dr Gillespie trained as a historian of science at the University of Melbourne and University of Pennsylvania. He taught history of science and medicine at the University of Pennsylvania, Deakin University and the University of Melbourne before joining the museum in 1990.
He has been a Fulbright Scholar (1980-1985), been awarded a Mellon Fellowship, Gordon Darling Fellowship, and received the Henry Schuman Prize of the History of Science Society (US).
He is a research fellow in the School of Historical Studies at the University of Melbourne, and a member of the Editorial Board of ReCollections: Journal of the National Museum of Australia.