Medical image segmentation is a critical task in computer vision, traditionally addressed using CNN- and Transformer-based models. However, CNNs struggle with long-range dependencies, while Transformers face quadratic computational complexity. Recent advancements in state-space models (SSMs), particularly the Mamba architecture, offer a promising solution by efficiently modeling long-range interactions with linear computational complexity. This paper presents a clear explanation of Mamba's fundamentals, detailing the transition from classical SSMs to the S6 model at its core. It also explores adaptations of Mamba for computer vision tasks, focusing on essential improvements and techniques for efficient training and inference while maintaining linear complexity.To help keep pace with the rapid progress of recent research into the application of Mamba in visual tasks, and considering the importance of medical image segmentation in the healthcare field, we take a closer look at the widespread applications of Mamba in 2D and 3D medical image segmentation,we classify these applications according to the type of medical data used and the procedure followed to utilize Mamba.Finally, we offer a critical discussion of the current state of the field, identifying key challenges and open issues, and describing promising future directions.