Artificial intelligence (AI) models for medical imaging tasks, such as classification or segmentation, require large and diverse datasets of images. However, due to privacy and ethical issues, as well as data sharing infrastructure barriers, these datasets are scarce and difficult to assemble. Synthetic medical imaging data generated by AI from existing data could address this challenge by augmenting and anonymizing real imaging data. In addition, synthetic data enable new applications, including modality translation, contrast synthesis, and professional training for radiologists. However, the use of synthetic data also poses technical and ethical challenges. These challenges include ensuring the realism and diversity of the synthesized images while keeping data unidentifiable, evaluating the performance and generalizability of models trained on synthetic data, and high computational costs. Since existing regulations are not sufficient to guarantee the safe and ethical use of synthetic images, it becomes evident that updated laws and more rigorous oversight are needed. Regulatory bodies, physicians, and AI developers should collaborate to develop, maintain, and continually refine best practices for synthetic data. This review aims to provide an overview of the current knowledge of synthetic data in medical imaging and highlights current key challenges in the field to guide future research and development.