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This paper introduces DextAIRity, an approach to manipulate deformable objects using active airflow. In contrast to conventional contact-based quasi-static manipulations, DextAIRity allows the system to apply dense forces on out-of-contact surfaces, expands the system's reach range, and provides safe high-speed interactions. These properties are particularly advantageous when manipulating under-actuated deformable objects with large surface areas or volumes. We demonstrate the effectiveness of DextAIRity through two challenging deformable object manipulation tasks: cloth unfolding and bag opening. We present a self-supervised learning framework that learns to effectively perform a target task through a sequence of grasping or air-based blowing actions. By using a closed-loop formulation for blowing, the system continuously adjusts its blowing direction based on visual feedback in a way that is robust to the highly stochastic dynamics. We deploy our algorithm on a real-world three-arm system and present evidence suggesting that DextAIRity can improve system efficiency for challenging deformable manipulation tasks, such as cloth unfolding, and enable new applications that are impractical to solve with quasi-static contact-based manipulations (e.g., bag opening). Video is available at https://youtu.be/_B0TpAa5tVo