He attended Queen Elizabeth's Grammar School, Kirkby Lonsdale, and won a scholarship to Christ's College, Cambridge.
[1] Burrow is best known for his thirty-two year tenure as Boden Professor of Sanskrit at the Oriental Institute of the University of Oxford (1944–1976).
His professional colleagues during this time included especially Richard Gombrich, Lecturer in Sanskrit from 1965, who would succeed him in the Boden Chair.
In these years he especially worked on the grammar and literature of the Niya Prakrit documents from Central Asia preserved in the Kharoṣṭhī script.
With his American colleague Murray B. Emeneau, Burrow published a milestone study, A Dravidian Etymological Dictionary (1966).