The exponential increase in video content poses significant challenges in terms of efficient navigation, search, and retrieval, thus requiring advanced video summarization techniques. Existing video summarization methods, which heavily rely on visual features and temporal dynamics, often fail to capture the semantics of video content, resulting in incomplete or incoherent summaries. To tackle the challenge, we propose a new video summarization framework that leverages the capabilities of recent Large Language Models (LLMs), expecting that the knowledge learned from massive data enables LLMs to evaluate video frames in a manner that better aligns with diverse semantics and human judgments, effectively addressing the inherent subjectivity in defining keyframes. Our method, dubbed LLM-based Video Summarization (LLMVS), translates video frames into a sequence of captions using an image caption model and then assesses the importance of each frame using an LLM, based on the captions in its local context. These local importance scores are refined through a global attention mechanism in the entire context of video captions, ensuring that our summaries effectively reflect both the details and the overarching narrative. Our experimental results demonstrate the superiority of the proposed method over existing ones in standard benchmarks, highlighting the potential of LLMs in the processing of multimedia content.