He was in Portugal when he passed away peacefully in his sleep. His family referred to him as a "Truly great man".
Croft's military sitcoms It Ain't Half Hot Mum and Dad's Army, written with Jimmy Perry, were hits in the 1970s.
He is also credited with Are You Being Served and its 1990s spin-off Grace and Favour.
Actor Melvyn Hayes, one of the stars of It Ain't Half Hot Mum, called Croft a "genius" and said it was "a privilege to work with" him.
"There were no swear words in his shows. His programmes were the kind of thing you could sit in front of the TV and watch with your grandmother and grandchildren," he told the BBC.
Thankyou for all of the entertainment and joy you brought to our screens. May you Rest In Peace David.
