His father was an accountant and he was educated at Raynes Park Grammar School and Exeter College, Oxford. He then became a journalist for the Sunday Chronicle (TV columnist), the Sunday Graphic (film and theatre columnist), the Sunday Times (radio critic and editor of Atticus) and the Sunday Telegraph (film critic).

He began working on television as a journalist in 1955. During the 1960s and 1970s, he presented the series Open House, Picture Parade, Points of View, Ask the Family, BBC3 – including the discussion during which Kenneth Tynan became the first person to say "fuck" on British television – and Call My Bluff. In 1967 it was Robinson who presented the edition of 'The Look of the Week' in which classical musicologist Hans Keller was brought face to face with the young Pink Floyd. He wrote and presented The Fifties on BBC1.Robinson was the presenter of The Book Programme on BBC2 from 1973-80 and a number of spin off documentaries, notably B. Traven - A Mystery Solved (1979). He wrote and presented several BBC1 documentaries under the title Robinson's Travels, among them The Mormon Trail (1976), Cruising and Indian Journey. In 1986 he wrote and presented The Magic Rectangle, one of the BBC1 documentaries marking the fiftieth anniversary of television.He also presented Today, BBC Radio 4's flagship morning news show, and Stop The Week, a fiercely competitive talk programme. Robinson fronted Brain of Britain on BBC Radio 4, but was replaced by Russell Davies during the 2004 series owing to illness. He returned to host the new series in 2005 until handing over the reins to Peter Snow in 2007. In September 2008 Robinson chaired the special Brain of Brains and Top Brain editions of the quiz and returned to host the series in 2008; Davies then replaced him for the 2009 shows.Paul Donovan "[Brain of Britain: A precious relic"], The Times, 4 October 2009 In August 2010 it was announced that Robinson is to permanently step down from Brain of Britain to be replaced by Davies.

Private Eye used to lampoon Robinson under the nickname 'Smuggins'. In a sketch on the BBC's Not the Nine O'Clock News he was impersonated by an actor wearing a cricket box over his forehead. Robinson has also been the subject of a sketch by Stephen Fry and Hugh Laurie in the second series of A Bit of Fry and Laurie, and Fry occasionally does an affectionate impression of Robinson when hosting the quiz show QI. He has also been lampooned by comedy duo David Mitchell and Robert Webb in the second series of That Mitchell and Webb Look, where he was shown as the presenter of an early version of their fictional gameshow Numberwang. He appeared in a Viz comic strip under the name Robin Robertson.

He is the father of the actress Lucy Robinson.