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Despite advances in data augmentation and transfer learning, convolutional neural networks (CNNs) difficultly generalise to unseen domains. When segmenting brain scans, CNNs are highly sensitive to changes in resolution and contrast: even within the same MRI modality, performance can decrease across datasets. Here we introduce SynthSeg, the first segmentation CNN robust against changes in contrast and resolution. SynthSeg is trained with synthetic data sampled from a generative model conditioned on segmentations. Crucially, we adopt a domain randomisation strategy where we fully randomise the contrast and resolution of the synthetic training data. Consequently, SynthSeg can segment real scans from a wide range of target domains without retraining or fine-tuning, which enables straightforward analysis of huge amounts of heterogeneous clinical data. Because SynthSeg only requires segmentations to be trained (no images), it can learn from labels obtained by automated methods on diverse populations (e.g., ageing and diseased), thus achieving robustness to a wide range of morphological variability. We demonstrate SynthSeg on 5,000 scans of six modalities (including CT) and ten resolutions, where it exhibits unparallelled generalisation compared with supervised CNNs, state-of-the-art domain adaptation, and Bayesian segmentation. Finally, we demonstrate the generalisability of SynthSeg by applying it to cardiac MRI and CT scans., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 The Author(s). Published by Elsevier B.V. All rights reserved.)