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Image quality assessment (IQA) is indispensable in clinical practice to ensure high standards, as well as in the development stage of machine learning algorithms that operate on medical images. The popular full reference (FR) IQA measures PSNR and SSIM are known and tested for working successfully in many natural imaging tasks, but discrepancies in medical scenarios have been reported in the literature, highlighting the gap between development and actual clinical application. Such inconsistencies are not surprising, as medical images have very different properties than natural images, and PSNR and SSIM have neither been targeted nor properly tested for medical images. This may cause unforeseen problems in clinical applications due to wrong judgment of novel methods. This paper provides a structured and comprehensive overview of examples where PSNR and SSIM prove to be unsuitable for the assessment of novel algorithms using different kinds of medical images, including real-world MRI, CT, OCT, X-Ray, digital pathology and photoacoustic imaging data. Therefore, improvement is urgently needed in particular in this era of AI to increase reliability and explainability in machine learning for medical imaging and beyond. Lastly, we will provide ideas for future research as well as suggesting guidelines for the usage of FR-IQA measures applied to medical images.