He was educated at Merchant Taylors' School and St John's College, Oxford, where he read Classics, taking Firsts in both Honour Moderations and literae humaniores and was elected to the Craven University Fellowship.

He participated in the excavations at Karanis organized by the University of Michigan, and published some Biblical papyri in the collections of the John Rylands Library.

[1] In 1954 Roberts succeeded A. L. P. Norrington as Secretary to the Delegates of Oxford University Press, holding the post until 1974.

In 1960 he held the Sandars Readership in Bibliography at Cambridge University lecturing on "The earliest manuscripts of the Church: style and significance."

Roberts was elected a Fellow of the British Academy in 1947, but resigned (along with his friend T. C. Skeat) in 1979, in protest against its decision not to expel the traitor Anthony Blunt from the fellowship.