The University of Kentucky has never been a 'recognized' national champion however they were selected as national champions in 1950 by the Sagarin Ratings and Bill Libby polls. They went 11-1 in 1950 under head coach Bear Bryant and defeated Oklahoma in the Sugar Bowl. Their only loss that season was to Tennessee who was named the national champion.
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Duke has never beaten Kentucky in a National Championship game. Kentucky did beat Duke for the title in 1978 (94-88). The game you are probably remembering is the 1992 East Regional Final won by Duke over Kentucky in overtime (104-103). Kentucky had taken the lead on Sean Woods' shot with 2.2 seconds left, then Duke's Grant Hill inbounded the ball 75 feet to Christian Laettner who spun around and hit the game winner as time ran out. This was the greatest game ever played in my humble opinion. Duke went on to beat Indiana (81-78) in the Final Four and Michigan (71-51) for their second consecutive national title.
Kentucky was in the NIT last in 2009. They lost to Notre Dame by 10 points in the quarterfinals.