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Quality assessment is a key element for the evaluation of hardware and software involved in image and video acquisition, processing, and visualization. In the medical field, user-based quality assessment is still considered more reliable than objective methods, which allow the implementation of automated and more efficient solutions. Regardless of increasing research in this topic in the last decade, defining quality standards for medical content remains a non-trivial task, as the focus should be on the diagnostic value assessed from expert viewers rather than the perceived quality from na\"{i}ve viewers, and objective quality metrics should aim at estimating the first rather than the latter. In this paper, we present a survey of methodologies used for the objective quality assessment of medical images and videos, dividing them into visual quality-based and task-based approaches. Visual quality based methods compute a quality index directly from visual attributes, while task-based methods, being increasingly explored, measure the impact of quality impairments on the performance of a specific task. A discussion on the limitations of state-of-the-art research on this topic is also provided, along with future challenges to be addressed.