Robert Peck was born into a working-class family in Leeds, West Riding of Yorkshire, on 23 August 1945. He went to Leeds Modern School in Lawnswood. Peck was educated at the Leeds College of Art where he received a Diploma in Art and Design.

After gaining popularity for his starring role in Edge of Darkness, Peck had become a national favourite and began appearing in films. After a few theatre appearances, Peck made his first appearance as a main character in a film again playing a policeman, John Graham, based in Kenya 1950, who takes under his wing the son of a murdered black priest in the 1987 film The Kitchen Toto. He also appeared in the 1987 film On the Black Hill, adapted from the 1982 novel of the same name by Bruce Chatwin. However, the film role that really launched his career as a film actor was his portrayal of the android Byron in the 1989 post-apocalyptic science-fiction adventure film Slipstream, in which he appeared alongside other big names including Mark Hamill, Bill Paxton, F. Murray Abraham, Ben Kingsley and Robbie Coltrane. During the late-1980s he also appeared in television shows including The Storyteller, The Jim Henson Hour and Screen One.

Peck's television career began in the 1970s, with his first television roles being in the BBC's Thirty-Minute Theatre anthology series in 1972, in which he appeared in the episode "Bypass". He also appeared in various other successful television productions such as Z-Cars and Play for Today. He also appeared in the films Royal Flash and Parker.

Before breaking into film and television work, Peck was a regular actor with the Royal Shakespeare Company alongside Ian McKellen, Donald Sinden and Judi Dench. Between 1979–80 he played Iago alongside Donald Sinden in Othello, in both Stratford and London. He made a memorable appearance on stage in the Royal Shakespeare Company production of The Life and Adventures of Nicholas Nickleby, originally by Charles Dickens, playing two characters: the boisterous Yorkshireman John Browdie and the predatory Sir Mulberry Hawk, and repeated these roles on Broadway and when the production was filmed for television in 1981. He played the character of Macduff in the Trevor Nunn's acclaimed 1976 stage and television versions of william Shakespeare's Macbeth, and re-appeared in another production of Macbeth in 1982. According to Peck's fellow Royal Shakespeare Theatre performer and veteran actor Sir Ian McKellen, Peck is the actor he considers he "learned the most from".

Peck married Actress Jill Baker in 1982, and they had three children Hannah (born 1983), George (born 1986) and Milly (born 1990). Peck and Baker shared a seventeen-year marriage until his death in 1999.

The series was broadcast on BBC Two in six episodes from 4 November to 9 December 1985. During its run the show attracted four million viewers and spiralled Peck to fame, winning him a British Academy Television Award for Best Actor at the British Academy of Film and Television Arts Awards. After the series ended, Peck had become a figure of popularity and a national favourite.

Peck won a British Academy Television Award for Best Actor at the British Academy of Film and Television Arts in 1986 for his portrayal of maverick policeman Ronald Craven in the television miniseries Edge of Darkness.

Peck's image and popularity increased with appearances in films including the 1990 film Lord of the Flies as the Marine Officer. He also appeared in the television shows Screen Two, Screenplay and A TV Dante. He also appeared in the television movies The Black Velvet Gown and An Ungentlemanly Act. In 1993, Peck made his biggest film appearance, when he was cast as park gamekeeper Robert Muldoon in the blockbuster smash hit Jurassic Park, directed by Steven Spielberg from the 1990 novel of the same name by Michael Crichton.

After appearing in Jurassic Park, Peck appeared in the popular television show The Young Indiana Jones Chronicles in 1993 playing General Targo in one episode. He also played Italian Romantic Composer Giuseppe Verdi in a television movie documenting the life and success of the Composer.

In the later years of his life and career, Peck appeared in more films portraying the roles of Captain Sebastian Belger in Merisairas, Françoise's father in Surviving Picasso, Ravn in Smilla's Sense of Snow, Harry Briggs in FairyTale: A True Story and Denton (based on Lancelot Dent) in the film The Opium War (Chinese name Yapian zhangzheng). He also appeared in the direct-to-TV film The Scold's Bridle (1998).

In November 1994, Peck was diagnosed with an undisclosed type of cancer. Peck was said to be undergoing chemotherapy and radiotherapy and his agent claimed that he was making a recovery. Nevertheless, Peck died at his home in London on 4 April 1999, at the age of 53. Peck's funeral took place in London, and his close friend and Edge of Darkness co-star Ian McNeice read a eulogy at the Service. He was cremated in London and his ashes were given to his family. He was survived by his widow and three children.

In 2000, a year after Peck's death from cancer, the stop-motion animated film The Miracle Maker, was released, in which Peck voiced the character of Joseph of Arimathea. The film was dedicated to Peck's memory.