With the rapid growth of online video platforms and the escalating volume of video content, the need for proficient video understanding tools has increased significantly. Given the remarkable capabilities of large language models (LLMs) in language and multimodal tasks, this survey provides a detailed overview of recent advances in video understanding that harness the power of LLMs (Vid-LLMs). The emergent capabilities of Vid-LLMs are surprisingly advanced, particularly their ability for open-ended multi-granularity (abstract, temporal, and spatiotemporal) reasoning combined with common-sense knowledge, suggesting a promising path for future video understanding. We examine the unique characteristics and capabilities of Vid-LLMs, categorizing the approaches into three main types: Video Analyzer × LLM, Video Embedder × LLM, and (Analyzer + Embedder) × LLM. We identify five subtypes based on the functions of LLMs in Vid-LLMs: LLM as Summarizer, LLM as Manager, LLM as Text Decoder, LLM as Regressor, and LLM as Hidden Layer. This survey also presents a comprehensive study of the tasks, datasets, benchmarks, and evaluation methods for Vid-LLMs. Additionally, it explores the extensive applications of Vid-LLMs in various domains, highlighting their remarkable scalability and versatility in real-world video understanding challenges. Additionally, it summarizes the limitations of existing Vid-LLMs and outlines directions for future research. For more information, readers are encouraged to visit the repository at https://github.com/yunlong10/Awesome-LLMs-for-Video-Understanding.