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On Wednesday, however, the Smash actress, 42, finally shared her little girl's given name -- and it's definitely noteworthy.
"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 actress’s rep Gabrielle Kachman tells People.
Unlike fellow celebs with interestingly named children, Thurman will not be elaborating about the origins behind baby Luna's name, but her rep does explain: "Each name has a special reason and meaning to her mother and father."
Thurman, who recently received Elle's Women in Hollywood honor from her congratulatory Bel Ami costar (and onscreen husband) Robert Pattinson Monday night.
Pattinson claimed that like his character, Thurman was "someone more intelligent, more charming, wittier and better in bed than me."
The award-winning actress and her French financier partner have five children between them: Thurman's two children with ex-husband Ethan Hawke -- Maya Ray and Levon -- and Busson's two sons from his relationship to Elle MacPherson -- Flynn and Cy.