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In this research, we successfully demonstrated dexterous manipulation of sheet-like elastic objects, namely, playing cards, using a high-speed robot system. In particular, our goal was to achieve card flicking by vibrating a fingertip of a robot hand at high speed and card catching by using high-speed visual feedback based on a high-speed vision system. We discuss card grasping in the initial state based on the geometry conditions of the card and the kinematics of the robot hand, and we propose a strategy for card flicking. We also suggest a card catching method based on information obtained by the high-speed vision system. In addition, we obtain the card flicking conditions by analyzing the slip between the card and the fingertip of the robot hand. Finally, we show experimental results of card flicking and card catching.