We introduce a new task called Referring Atomic Video Action Recognition (RAVAR), aimed at identifying atomic actions of a particular person based on a textual description and the video data of this person. This task differs from traditional action recognition and localization, where predictions are delivered for all present individuals. In contrast, we focus on recognizing the correct atomic action of a specific individual, guided by text. To explore this task, we present the RefAVA dataset, containing 36,630 instances with manually annotated textual descriptions of the individuals. To establish a strong initial benchmark, we implement and validate baselines from various domains, e.g., atomic action localization, video question answering, and text-video retrieval. Since these existing methods underperform on RAVAR, we introduce RefAtomNet -- a novel cross-stream attention-driven method specialized for the unique challenges of RAVAR: the need to interpret a textual referring expression for the targeted individual, utilize this reference to guide the spatial localization and harvest the prediction of the atomic actions for the referring person. The key ingredients are: (1) a multi-stream architecture that connects video, text, and a new location-semantic stream, and (2) cross-stream agent attention fusion and agent token fusion which amplify the most relevant information across these streams and consistently surpasses standard attention-based fusion on RAVAR. Extensive experiments demonstrate the effectiveness of RefAtomNet and its building blocks for recognizing the action of the described individual. The dataset and code will be made publicly available at https://github.com/KPeng9510/RAVAR.