But the monicker of the little one has never been revealed… until now - and it's certainly going onto the list of unusual celebrity baby names.
"I would like to announce Uma and Arki's daughter's name for the first time officially: Rosalind Arusha Arkadina Altalune Florence Thurman-Busson, better known to family and friends as Luna," the star's rep told People.
Uma and Arpad didn't elaborate on exactly what each name meant but her spokesperson said that "each name has a special reason and meaning to her mother and father."
Luna is the first child for the Kill Bill actress and her beau although Uma has two children; Maya and Levon, from her marriage to Ethan Hawke.
Arpad has two sons; Flynn and Cy with ex-fiancee, supermodel Elle MacPherson.
What do you think of Uma's daughter's name?