The writer died of natural causes at his home on Saturday night, according to a statement released by his publicist Cynthia Swartz.
Frank achieved his first big success with the play The Subject Was Roses, which won the playwright a Pulitzer Prize, a Tony Award and a Drama Circle honour.
The play was later adapted into a film starring Martin Sheen, Patricia Neal and Jack Albertson in 1968 and received two Oscar nominations, with Albertson winning the Best Supporting Actor award.
Among his other filmmaking credits, Frank wrote 1956's The Fastest Gun Alive starring Glenn Ford, as well as The Only Game in Town which featured Elizabeth Taylor and Warren Beatty in 1969.
Frank's sons followed their father into Hollywood, with Tony being the screenwriter behind the Matt Damon-starring Bourne films, and Dan directing Jake Gyllenhaal in 2014's Nightcrawler. John Gilroy also works as an editor, whose recent work includes Nightcrawler and The Bourne Legacy.
Frank is survived by his wife Ruth, his three sons John, Tony and Dan, and his five grandchildren.