Gascoigne was born in London and won scholarships to both Eton College and Magdalene College, Cambridge, where he studied English Literature. He then spent a year as a Commonwealth Fund Scholar at Yale University (1958-59). After National Service in the Grenadier Guards he became a theatre critic.

He came to fame as the presenter of the popular television quiz show, University Challenge, in 1962, based on the US series College Bowl. Although he has written several books, mostly on history, and presented other television programmes, his name is permanently connected with University Challenge in the minds of many people.

His questioning manner was firm yet polite, and his judgment scrupulously fair. Phrases he often used which became catchphrase include: "Your starter for ten", "fingers on the buzzers" and "I'll have to hurry you".

He returned in 1998 to present the Red Dwarf special Universe Challenge, a contest between the actors of the show and a set of fans; the fans won by a narrow margin.

In 1977, Granada Television broadcast on ITV The Christians, a 13 x 1 hour television documentary series on the history of Christianity, written and presented by Gascoigne. He and his wife Christina, a photographer, simultaneously published a book with the same title, telling the story in printed form.

In 1987, Gascoigne presented a documentary series of 6 x 30 minute programmes on Victorian history, Victorian Values, which looked at how Victorian society put in place the infrastructure of the modern welfare state, also produced by Granada Television.

Another TV series that he presented was The Great Moghuls which is a tour of the Moghul (or more correctly Mughal) dynasty of India. The series was accompanied by a book whose soft-cover edition has been recently re-issued.

In recent years he has devoted much of his time in establishing an online history encyclopaedia called History World.

In the early hours of Wednesday 8 August 1979 Gascoigne was witness to the burial by Kit Williams, the author of Masquerade, of a valuable golden hare in an earthenware jar "somewhere in Britain". The treasure hunt which followed was pursued worldwide and chronicled in the book Quest for the Golden Hare.

In the Young Ones episode "Bambi" he is parodied by Griff Rhys Jones as "Bambi Gascoigne" (with considerable emphasis being placed on the resemblance of his name to the Disney character), and several years previously Griff Rhys Jones played Gascoigne in a sketch on Not the Nine O'Clock News. He was also portrayed by actor Mark Gatiss in the film Starter for 10 (2006).

In 2004 he appeared as a television presenter in an episode of Jonathan Creek, "Gorgon's Wood".

His name appears in one version of the Monty Python "Lumberjack Song" when Michael Palin sings of the "Quercus maximus Bamber Gascoigneii", and in the Python song "I Like Traffic Lights" the singer points out that his name is not Bamber.

His latest project is TimeSearch which presents a number of timelines.