Medical image segmentation plays a critical role in computer-assisted diagnosis, treatment planning, surgical navigation, and quantitative medical image analysis. Among deep learning-based segmentation architectures, U-Net has become one of the most influential frameworks because of its encoder–decoder structure and skip connections for integrating semantic and spatial information. This review focuses on U-Net and representative variants for two-dimensional medical image segmentation. We systematically analyze their architectural evolution, including attention-based designs, enhanced skip connections, deformable convolutions, and Transformer-based extensions and discuss their advantages, limitations, and application boundaries. Quantitative performance results are synthesized from published studies rather than reproduced through new benchmarking experiments; direct numerical comparisons are restricted to models evaluated within the same study or under clearly defined benchmark settings. We further relate architectural modifications to modality-specific challenges in MRI, CT, ultrasound, fundus imaging, dermoscopy, and microscopy. Finally, current challenges in computational efficiency, cross-domain generalization, data scarcity, interpretability, and clinical deployment are discussed, together with promising directions for future research.