New York Daily News was first to report that news, stating that Arpad, 51, filed the action in Manhattan Supreme Court.
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Arpad Busson and Uma Thurman at the 2013 Cannes Film Festival
Details about the proceeding are not immediately clear; typically custody battles are fought in matrimonial court, where records are automatically sealed.
The case has been assigned to Manhattan Supreme Court Justice Matthew Cooper, who handles divorce, custody and support cases full time.
Uma, 44, and Arpad started dating in 2007, two years after her divorce from Ethan Hawke. The couple, who were engaged to be married, welcomed their daughter Luna in July, 2012. They ended their relationship in April.
Uma Thurman and Arpad Busson called off their engagement in April 2014
The actress has two other children from her marriage to Ethan; Maya, 16, and 12-year-old Levon.
Arpad, meanwhile, has a 16-year-old son Flynn and 11-year-old son Aurelius from his previous relationship with Elle Macpherson.