Large Language Models (LLMs) are transforming AI across sectors like healthcare, robotics, and IoT, yet the limitations of cloud-based deployments regarding latency, bandwidth, and privacy are driving a shift toward edge computing. By processing data closer to the source, edge deployments enhance real-time processing, privacy, and operational efficiency. However, deploying LLMs at the edge poses challenges due to the resource-intensive nature of these models, such as high computational demands, memory and storage constraints, and energy efficiency issues. Innovative solutions like model quantization, parameter-efficient fine-tuning, and distributed computing are being developed to optimize edge deployments. Architectural innovations, including task-oriented designs and edge model caching, aim to enhance performance while maintaining privacy and scalability. The current landscape sees edge-based LLMs reshaping industries by enabling more localized and efficient AI applications, particularly in healthcare, robotics, IoT networks, and autonomous driving, while future advancements focus on overcoming resource constraints, ensuring energy efficiency, and enhancing privacy.