Estimating human gaze targets from images in-the-wild is an important and formidable task. Existing approaches primarily employ brittle, multi-stage pipelines that require explicit inputs, like head bounding boxes and human pose, in order to identify the subject of gaze analysis. As a result, detection errors can cascade and lead to failure. Moreover, these prior works lack the flexibility of specifying the gaze analysis task via natural language prompting, an approach which has been shown to have significant benefits in convenience and scalability for other image analysis tasks. To overcome these limitations, we introduce the Promptable Gaze Target Estimation (PGE) task, a new end-to-end, concept-driven paradigm for gaze analysis. PGE conditions gaze prediction on flexible user text or visual prompts (e.g., "the boy in the red shirt" or "person in point [0.52, 0.48]") to identify a specific subject for gaze analysis. This approach integrates subject localization with gaze estimation, and eliminates the rigid dependency on intermediate analysis stages. We develop a scalable data engine to generate Gaze-Co (Gaze Estimation with Concepts), a dataset and benchmark of 120K high-quality, prompt-annotated image pairs. We also propose GazeAnywhere, the first model designed for PGE. GazeAnywhere uses a transformer-based detector to fuse features from frozen encoders and simultaneously solves subject localization, in/out-of-frame presence, and gaze target heatmap estimation. GazeAnywhere achieves state-of-the-art performance on multiple PGE benchmarks, setting a strong baseline for this new problem even on a difficult out-of-domain, real-world clinical dataset. GazeAnywhere is open-sourced in github.com/IrohXu/GazeAnywhere.