Robert was born into the family of the British television director John Glenister (born John Glenister, born October 12, 1932). He has a younger brother, Philip Haywood Glenister (born Philip Haywood Glenister, born February 10, 1963), also an actor. From 1984 to 1992, he was married to the English actress Amanda Redman. From this marriage, he has a daughter, Emily Joanna Glenister (born Emily Joanna Glenister, born July 1, 1987), a television actress. From his second wife, Celia Glenister, he has a son, Thomas Glenister.