Educated at Sedbergh School, then in Yorkshire, Wright specialised in Classics.
From 1968 to 1971, he studied Literae Humaniores (or “Classics”, i.e. classical literature, philosophy, and history) at Exeter College, Oxford, receiving his BA with First Class Honours in 1971.
During that time he was president of the undergraduate Oxford Inter-Collegiate Christian Union. In 1973 he received a BA in theology with First Class Honours from Exeter.
From 1971 to 1975 he studied for the Anglican ministry at Wycliffe Hall, Oxford, receiving his (Oxford) MA at the end of this period.
In 1975 he became a junior research fellow at Merton College, Oxford and later also junior chaplain. From 1978 to 1981 he was a fellow and Chaplain at Downing College, Cambridge. In 1981 he received his DPhil from Merton College, Oxford, his thesis topic being “The Messiah and the People of God: A Study in Pauline Theology with Particular Reference to the Argument of the Epistle to the Romans”.
After this, he served as assistant professor of New Testament Studies at McGill University, Montreal (1981–1986), then as Chaplain, Fellow and Tutor at Worcester College and lecturer in New Testament in the University of Oxford (1986–1993).
He moved from Oxford to be Dean of Lichfield Cathedral (1994–1999) and then returned briefly to Oxford as Visiting Fellow of Merton College, before taking up his appointment as Canon Theologian of Westminster Abbey in 2000.
In 2003, he became the Bishop of Durham.
On 4 August 2006 he was appointed to the Court of Ecclesiastical Causes Reserved for a period of five years.
On 27 April 2010 it was announced that he will be retiring from the See of Durham on 31 August 2010 to take up a new appointment as Research Professor of New Testament and Early Christianity at the University of St Andrews in Scotland, which will enable him to concentrate on his academic and broadcasting work.