In recent years, the segmentation of anatomical or pathological structures using deep learning has experienced a widespread interest in medical image analysis. Remarkably successful performance has been reported in many imaging modalities and for a variety of clinical contexts to support clinicians in computer-assisted diagnosis, therapy, or surgical planning purposes. However, despite the increasing amount of medical image segmentation challenges, there remains little consensus on which methodology performs best. Therefore, we examine in this article the numerous developments and breakthroughs brought since the rise of U-Net-inspired architectures. Especially, we focus on the technical challenges and emerging trends that the community is now focusing on, including conditional generative adversarial and cascaded networks, medical Transformers, contrastive learning, knowledge distillation, active learning, prior knowledge embedding, cross-modality learning, multistructure analysis, federated learning, or semi-supervised and self-supervised paradigms. We also suggest possible avenues to be further investigated in future research efforts.