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This paper explores the use of deep learning-based computer vision for real-time monitoring of the flow in intravenous (IV) infusions. IV infusions are among the most common therapies in hospitalized patients and, given that both over-infusion and under-infusion can cause severe damages, monitoring the flow rate of the fluid being administered to patients is very important for their safety. The proposed system uses a camera to film the IV drip infusion kit and a deep learning-based algorithm to classify acquired frames into two different states: frames with a drop that has just begun to take shape and frames with a well-formed drop. The alternation of these two states is used to count drops and derive a measurement of the flow rate of the drip. The usage of a camera as sensing element makes the proposed system safe in medical environments and easier to be integrated into current health facilities. Experimental results are reported in the paper that confirm the accuracy of the system and its capability to produce real-time estimates. The proposed method can be therefore effectively adopted to implement IV infusion monitoring and control systems.