Norman was educated at Chatham House Grammar School, Ramsgate, Kent. He won an Open Scholarship to Selwyn College, Cambridge.
He was a Fellow of Selwyn College, Cambridge from 1962 to 1964, before moving to Jesus College as a Fellow. Norman lectured in history at the University of Cambridge; he is an emeritus Fellow of Peterhouse, Cambridge.
He was Dean of Peterhouse for seventeen years and Dean and Chaplain at Christ Church College, Canterbury and Professor of History at the University of York. He is a member of the Peterhouse school of history. On 7 October 2012, he was received into the Catholic Church by way of the Personal Ordinariate of Our Lady of Walsingham.
Norman was the BBC Reith Lecturer in 1978. For his series of six radio lectures, entitled "Christianity and the World", he discussed the relationship between religion and politics. Margaret Thatcher once invited him to Chequers, although Norman insists he is not a Thatcherite and says he is "appalled by the results of naked capitalism".
- The Catholic Church and Ireland (1965)
- The Conscience of the State in North America (1968)
- Anti-Catholicism in Victorian England (1968)
- The Early Development of Irish Society (1969)
- A History of Modern Ireland (1971)
- Church and Society in Modern England (1976)
- "Christianity and Politics" in Maurice Cowling (ed.), Conservative Essays (Cassell, 1978, pp. 69–81.)
- Christianity and the World BBC Reith Lectures (1978)
- Christianity and the World Order Book based on the BBC Reith Lectures (1979)
- Christianity in the Southern Hemisphere (1981)
- The English Catholic Church in the Nineteenth Century (1983)
- Roman Catholicism in England (1985)
- The Victorian Christian Socialists (1987)
- The House of God: Church Architecture, Style and History (1990)
- Entering the Darkness: Christianity and its modern substitutes (1991)
- An Anglican Catechism (2001)
- Out of the Depths (2001)
- Secularisation (2002)
- Anglican Difficulties (2004)
- The Mercy of God's Humility (2004)
- The Roman Catholic Church (2006)
- Secularisation, and The House of God: Church Architecture, Style and History (1990)