Healthcare professionals have been increasingly viewing medical images and videos in their routine clinical practice, and this in a wide variety of environments. Both the perception and interpretation of medical visual information, across all branches of practice or medical specialties (e.g. diagnostic, therapeutic, or surgical medicine), career stages, and practice settings (e.g. emergency care), appear to be critical for patient care. However, medical images and videos are not self-explanatory and, therefore, need to be interpreted by humans, i.e. medical experts. In addition, various types of degradations and artifacts may appear during image acquisition or processing, and consequently affect medical imaging data. Such distortions tend to impact viewers' quality of experience, as well as their clinical practice. It is accordingly essential to better understand how medical experts perceive the quality of visual content. Thankfully, progress has been made in the recent literature towards such understanding. In this article, we present an up-to-date state-of the-art of relatively recent (i.e. not older than ten years old) existing studies on the subjective quality assessment of medical images and videos, as well as research works using task-based approaches. Furthermore, we discuss the merits and drawbacks of the methodologies used, and we provide recommendations about experimental designs and statistical processes to evaluate the perception of medical images and videos for future studies, which could then be used to optimise the visual experience of image readers in real clinical practice. Finally, we tackle the issue of the lack of available annotated medical image and video quality databases, which appear to be indispensable for the development of new dedicated objective metrics.