Alan Turing was a pioneering computer scientist who helped crack the Enigma code during WWII, was convicted of gross indecency in 1952 for being gay, and took his own life two years later.
Turing was given a royal pardon by The Queen in 2013, but Cumberbatch and other campaigners want a pardon for 49,000 other men prosecuted because of their sexuality.
UK GAY LAWS: A TIMELINE
- 1533: Under the Buggery Act, same-sex activity was banned and punishable with the death penalty
- 1861 The Offences Against the Person Act formally abolished the death penalty for buggery (anal sex, oral sex or sex with an animal) in England and Wales
- 1967: Gay sexual activity is decriminalised in England and Wales under the Sexual Offences Act, as long as it is consensual, "in private" and involves adults over 21
- 1981: Gay sex becomes legal in Scotland under the same terms as England and Wales. Northern Ireland follows in 1982
- 1994: House of Commons votes to lower the age of consent for gay people to 18
- 1999: First protection against discrimination for gay people comes into law
- 2000: Ban on LGBT people serving openly in the armed forces was lifted
- 2001: Age of consent for gay people lowered from to 16, the same as for straight people
- 2014: Same-sex marriage legalised in England, Wales and Scotland
Turing, whom Winston Churchill described as having "made the single biggest contribution to the Allied victory in the Second World War" was forced to choose between 'treatment' and imprisonment, and underwent hormonal injections which left him impotent.
The Queen granted the scientist a posthumous pardon under the Royal Prerogative of Mercy in 2013, but the campaigners say this should be extended to all men who fell foul of the law.
In a letter to the Government and published in The Guardian, they are calling for the royal family to act and convince the Government to pardon all those convicted under the outdated law.
The letter states: "The UK's homophobic laws made the lives of generations of gay and bisexual men intolerable.
"It is up to young leaders of today including the Duke and Duchess of Cambridge to acknowledge this mark on our history and not allow it to stand.
"We call upon Her Majesty's Government to begin a discussion about the possibility of a pardoning all the men, alive or deceased, who like Alan Turing, were convicted."
A spokesman for the Duke and Duchess of Cambridge said that as this is a matter for government they would not make any public comment on the issue.
Stephen Fry, who also signed the campaign's letter, married his partner Elliot Spencer in January.
The letter is also signed by Morten Tyldum, director of The Imitation Game, Rachel Barnes who is Alan Turing's niece, and Matthew Todd, editor of Attitude Magazine.
In 2009, an "unequivocal apology" for Turing's appalling treatment was issued by then prime minister Gordon Brown.
The campaign states: "The apology and pardon of Alan Turing are to be welcomed but ignores over 49,000 men who were convicted under the same law, many of whom took their own lives.
"An estimated 15,000 men are believed to still be alive."
An online petition - at www.Pardon49k.org - has so far been signed by more than 69,000 people.