We propose a novel framework for video understanding, called Temporally Contextualized CLIP (TC-CLIP), which leverages essential temporal information through global interactions in a spatio-temporal domain within a video. To be specific, we introduce Temporal Contextualization (TC), a layer-wise temporal information infusion mechanism for videos, which 1) extracts core information from each frame, 2) connects relevant information across frames for the summarization into context tokens, and 3) leverages the context tokens for feature encoding. Furthermore, the Video-conditional Prompting (VP) module processes context tokens to generate informative prompts in the text modality. Extensive experiments in zero-shot, few-shot, base-to-novel, and fully-supervised action recognition validate the effectiveness of our model. Ablation studies for TC and VP support our design choices. Our project page with the source code is available at https://github.com/naver-ai/tc-clip