<p>The paradigm of large language models (LLMs) is undergoing a fundamental shift from centralized cloud to the network edge, driven by urgent needs for data privacy, low-latency interaction, and offline reliability. However, deploying resource-intensive LLMs on constrained edge devices, such as smartphones and vehicles, presents a significant technical challenge. This survey provides a holistic, end-to-end engineering perspective on edge LLMs, systematically charting the path from on-device adaptation to efficient inference. We begin with emerging fine-tuning techniques, such as parameter-efficient fine-tuning and federated learning, which are crucial for personalization. The core of this work is a deep dive into the three pillars of efficient inference: model compression (quantization, pruning, and distillation), optimized architectural designs (e.g., attention mechanism variants), and critical runtime and system-level optimizations (e.g., key-value cache management, speculative decoding, and memory paging). We further establish a framework for holistic evaluation and survey the burgeoning applications across diverse domains. Unlike existing reviews that may focus on narrower topics, such as security, specific device categories, or high-level future concepts, this survey serves as a practical, unified roadmap for researchers and practitioners. It bridges the gap between algorithmic theory and system-level implementation, providing a comprehensive guide to building the next generation of private, responsive, and intelligent edge LLM applications.</p>