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Recent advances in computer vision have shown promising results in image generation. Diffusion probabilistic models have generated realistic images from textual input, as demonstrated by DALL-E 2, Imagen, and Stable Diffusion. However, their use in medicine, where imaging data typically comprises three-dimensional volumes, has not been systematically evaluated. Synthetic images may play a crucial role in privacy-preserving artificial intelligence and can also be used to augment small datasets. We show that diffusion probabilistic models can synthesize high-quality medical data for magnetic resonance imaging (MRI) and computed tomography (CT). For quantitative evaluation, two radiologists rated the quality of the synthesized images regarding "realistic image appearance", "anatomical correctness", and "consistency between slices". Furthermore, we demonstrate that synthetic images can be used in self-supervised pre-training and improve the performance of breast segmentation models when data is scarce (Dice scores, 0.91 [without synthetic data], 0.95 [with synthetic data]).