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Father Addams is the deceased father of Gomez Addams, Uncle Fester, Pancho Addams and Cosimo Addams.
He was the husband of Grandmama, therefore making her his widow, but in his only appearance (as a statue), his wife is Mother Addams, since Grandmama was Morticia Addams' mother in that incarnation.
Charles Addams cartoons
A possible picture of him and a younger Grandmama can be seen on the cover of Afternoon in the Attic.
The 1964 series
Grandmama said he loved her Yak Gravy on Glutton Bread. When Morticia corrected her "That's gluten bread," she replied "Not the way he ate it." Eating too much of her Yak Stew made him sleep-walk.[1]
The Addams Family (1991)
Morticia mentions Mother and Father Addams as she walks in the cemetery with Fester. She wishes the children could've known them better: "...tell that to an angry mob."[2]
The Addams Family Values (1993)
When Gomez comments about Pubert Addams that he has Gomez' father's eyes, Morticia Addams tells him to 'take those out of his mouth', implying that he may have worn fake eyes and/or left them behind when he died.
Description
Father Addams died an elderly man, with a slight beard--presumably grey. He wore a top hat and a suit. It is mentioned in one of the incarnations that he started growing hair when he was 50 years old, which would mean that baldness is a trait in the family (and is passed on to Fester).