Shaw was born and raised in Belfast, one of three brothers. He originally trained as a structural engineer, but also worked as an aircraft designer and journalist. He and his first wife Sadie and their children left Northern Ireland and lived in Canada for a few years before settling in Ulverston, England in the mid-1970s, because they were worried about the political situation. They subsequently moved to Grappenhall in Warrington, where Shaw lived for some years after Sadie's death in 1991. Throughout this period they were members of the Liverpool Science Fiction Group (LiG). Shaw died of cancer on 11 February 1996. He was survived by one son and two daughters, and his second wife Nancy Tucker, who he had married in 1995.

Shaw is perhaps best known for "Light of Other Days" (August 1966), the story that introduced the concept of slow glass, through which the past can be seen. Shaw sold this story to Analog editor John W. Campbell, who liked it so much Shaw wrote a sequel for him, "Burden of Proof", in May 1967. The original story was written in four hours, but after years of planning. Shaw expanded on the concept in the novel Other Days, Other Eyes, and the concept was adopted by the Marvel Comics/Curtis Magazines anthology magazine Unknown Worlds of Science Fiction.

His work ranged from essentially mimetic stories with fantastic elements far in the background (Ground Zero Man) to van Vogtian extravaganzas (The Palace of Eternity). Later in his career he began writing trilogies: The Land trilogy (The Ragged Astronauts, The Wooden Spaceships and The Fugitive Worlds) was set on system of worlds where technology has evolved with no metals.

Most of Shaw's novels and short stories are serious, but he was known in the fan community for his wit. His early fanwriting career was as a member of Irish Fandom, aka the Wheels of IF, along with Walt Willis, and James White. He always remained a keen reader of and contributor to fanzines. Later, and for many years, at the British science fiction convention Eastercon, he would deliver a humorous speech (often part of his famous series known by the tongue-in-cheek label of "Serious Scientific Talks"); these were eventually collected in The Eastercon Speeches (1979) and A Load of Old Bosh (1995), which also included a similar talk from the 1979 Worldcon in Brighton, 37th World Science Fiction Convention. With Walt Willis, he was the author of the allegorical The Enchanted Duplicator, an oft-reprinted 1954 piece of fiction about science fiction fandom explicitly modeled on John Bunyan's The Pilgrim's Progress (though Shaw and Willis denied having read it beforehand ). He was also (with Brian Aldiss and Harry Harrison) co-president of the Birmingham Science Fiction Group.

• Night Walk (1967). London : Gollancz.
• The Two-Timers (1968). New York : Ace Books.
• The Palace Of Eternity (1969). New York : Ace Pub. Corp.
• The Shadow Of Heaven (1969). New York : Avon.
• One Million Tomorrows (1970). New York : Ace books.
• Ground Zero Man (1971). New York: Avon Books. - revised edition published as The Peace Machine (1985). London : Gollancz.
• Other Days, Other Eyes (1972). New York : Ace Books.
• Tomorrow Lies In Ambush (1973). New York : Ace Books. - collection.
• Tomorrow Lies In Ambush (1973). London: Gollancz. - collection.
• The Orbitsville Trilogy
• Orbitsville (1975). London : Gollancz.
• Orbitsville Departure (1983). New York : Daw Books.
• Orbitsville Judgement (1990). London : Gollancz.
• A Wreath of Stars (1976). London : Gollancz.
• Cosmic Kaleidoscope (1976). London: Gollancz. - collection.
• Cosmic Kaleidoscope (1977). New York: Doubleday - collection.
• Medusa's Children (1977). New York: Doubleday.
• The Warren Peace saga
• Who Goes Here? (1977). London : Gollancz. - reissued in 1988 with a short story The Giaconda Caper.
• Warren Peace (1993). London : Gollancz. - reissued in 1994 as Dimensions
• Ship Of Strangers (1978). London : Gollancz - collection.
• Vertigo (1978). London : Gollancz. reissued in 1991 as Terminal Velocity by the same publisher.
• Dagger of the Mind (1979). London : Gollancz.
• The Ceres Solution (1981). London : Granada.
• Galactic Tours (1981, with David A. Hardy).
• Courageous New planet (1981). Birmingham Science Fiction Group - limited-edition chapbook.
• A Better Mantrap (1982). London : Gollancz - collection.
• Fire Pattern (1984). London : Gollancz.
• Messages Found in an Oxygen Bottle (1986). Cambridge, Mass. : Nesfa. - collection. Bound double format with Between Two Worlds by Terry Carr
• Land and Overland Trilogy
• The Ragged Astronauts (1986). London : Gollancz.
• The Wooden Spaceships (1988). London : Gollancz.
• The Fugitive Worlds (1989). London : Gollancz.
• Killer Planet (1989). London : Gollancz.
• Dark Night In Toyland (1989). London : Gollancz - collection.
• Overload (1995). Birmingham Science Fiction Group - limited-edition chapbook.

Nonfiction

• The Best of the Bushel (1979)
• The Eastercon Speeches (1979)
• How to Write Science Fiction (1993)
• A Load of Old BoSh (1995) (includes The Eastercon Speeches)

Partial list of short stories

• "Light of Other Days" (1967)
• "Skirmish on a Summer Morning" (1976)
• "Unreasonable Facsimile" (1974)
• "A Full Member of the Club" (1974)
• "The Silent Partners" (1959)
• "The Element of Chance" (1969)
• "The Gioconda Caper" (1976)
• "An Uncomic Book Horror Story" (1975)
• "Deflation 2001" (1972)
• "Waltz of the Bodysnatchers" (1976)
• "A Little Night Flying" ("Dark Icarus") (1975)