Large language models (LLMs) have revolutionized natural language processing with their exceptional understanding, synthesizing, and reasoning capabilities. However, deploying LLMs on resource-constrained edge devices presents significant challenges due to computational limitations, memory constraints, and edge hardware heterogeneity. This survey provides a comprehensive overview of recent advancements in edge LLMs, covering the entire lifecycle: from resource-efficient model design and pre-deployment strategies to runtime inference optimizations. It also explores on-device applications across various domains. By synthesizing state-of-the-art techniques and identifying future research directions, this survey bridges the gap between the immense potential of LLMs and the constraints of edge computing.