Biological systems are three-dimensional and complex. Images of biological 3D structures can be acquired using different imaging technologies and at increasing resolutions. However, identifying relevant structural features in complex 3D-imaged subjects remains a significant challenge. 3D image segmentation is usually performed using deep-learning segmentation models. Such models are trained on manually annotated and segmented, dataset-specific images. Consequently, such models rarely generalize across datasets. Here, we overcome these limitations with TopoTome, an unsupervised 3D image segmentation and analysis algorithm based on topological data analysis. TopoTome encodes the 3D image directly in topological space. It then performs unsupervised clustering to detect and segment features represented by salient voxel intensity gradients. We demonstrate on simple, complex, synthetic and real-world 3D image data that it outperforms all 3D clustering algorithms. Its segmentation ranks with or outperforms best-in-class deep learning 3D image segmentation software. Besides segmentation, it enables streamlined topological data analysis of 3D images. Owing to its conceptually different topological data analysis core, TopoTome does not need prior information and tuning to generalize across different imaging modalities, including fluorescence microscopy and X-ray computed tomography. We show it also readily generalizes across biological subjects, such as different species, organs and cells. TopoTome is thus one of the most versatile and accurate unsupervised 3D image segmentation algorithms.