Kukura studied acting at the Academy of Performing Arts in Bratislava (VŠMU). Since 1985, when he emigrated to Germany, his films were banned by the regime until the end of Communist rule in the 1989. Thanks to his charismatic personality he usually portrayed strong, leading characters. In 2004, he received DOSKY Award for performing "Martin" in "The Goat, or Who is Sylvia?" by Edward Albee. In 2003 he became managing director of Arena Theatre.