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Medical image segmentation plays a crucial role in various healthcare applications, enabling accurate diagnosis, treatment planning, and disease monitoring. Traditionally, convolutional neural networks (CNNs) dominated this domain, excelling at local feature extraction. However, their limitations in capturing long-range dependencies across image regions pose challenges for segmenting complex, interconnected structures often encountered in medical data. In recent years, Vision Transformers (ViTs) have emerged as a promising technique for addressing the challenges in medical image segmentation. Their multi-scale attention mechanism enables effective modeling of long-range dependencies between distant structures, crucial for segmenting organs or lesions spanning the image. Additionally, ViTs' ability to discern subtle pattern heterogeneity allows for the precise delineation of intricate boundaries and edges, a critical aspect of accurate medical image segmentation. However, they do lack image-related inductive bias and translational invariance, potentially impacting their performance. Recently, researchers have come up with various ViT-based approaches that incorporate CNNs in their architectures, known as Hybrid Vision Transformers (HVTs) to capture local correlation in addition to the global information in the images. This survey paper provides a detailed review of the recent advancements in ViTs and HVTs for medical image segmentation. Along with the categorization of ViT and HVT-based medical image segmentation approaches, we also present a detailed overview of their real-time applications in several medical image modalities. This survey may serve as a valuable resource for researchers, healthcare practitioners, and students in understanding the state-of-the-art approaches for ViT-based medical image segmentation.