Uncrewed Aerial Vehicles (UAVs) are widely deployed across diverse applications due to their mobility and agility. Recent advances in Large Language Models (LLMs) offer a transformative opportunity to enhance UAV intelligence beyond conventional optimization-based and learning-based approaches. By integrating LLMs into UAV systems, advanced environmental understanding, swarm coordination, mobility optimization, and high-level task reasoning can be achieved, thereby allowing more adaptive and context-aware aerial operations. This survey systematically explores the intersection of LLMs and UAV technologies and proposes a unified framework that consolidates existing architectures, methodologies, and applications for UAVs. We first present a structured taxonomy of LLM adaptation techniques for UAVs, including pretraining, fine-tuning, Retrieval-Augmented Generation (RAG), and prompt engineering, along with key reasoning capabilities such as Chain-of-Thought (CoT) and In-Context Learning (ICL). We then examine LLM-assisted UAV communications and operations, covering navigation, mission planning, swarm control, safety, autonomy, and network management. After that, the survey further discusses Multimodal LLMs (MLLMs) for human-swarm interaction, perception-driven navigation, and collaborative control. Finally, we address ethical considerations, including bias, transparency, accountability, and Human-in-the-Loop (HITL) strategies, and outline future research directions. Overall, this work positions LLM-assisted UAVs as a foundation for intelligent and adaptive aerial systems.