This paper proposes Panoptic Narrative Grounding , a spatially fine and general formulation of the natural language visual grounding problem. We establish an experimental framework for the study of this new task, including new ground truth and metrics. We propose PiGLET, a novel multi-modal Transformer architecture to tackle the Panoptic Narrative Grounding task, and to serve as a stepping stone for future work. We exploit the intrinsic semantic richness in an image by including panoptic categories, and we approach visual grounding at a fine-grained level using segmentations. In terms of ground truth, we propose an algorithm to automatically transfer Localized Narratives annotations to specific regions in the panoptic segmentations of the MS COCO dataset. PiGLET achieves a performance of 63.2 absolute Average Recall points. By leveraging the rich language information on the Panoptic Narrative Grounding benchmark on MS COCO, PiGLET obtains an improvement of 0.4 Panoptic Quality points over its base method on the panoptic segmentation task. Finally, we demonstrate the generalizability of our method to other natural language visual grounding problems such as Referring Expression Segmentation. PiGLET is competitive with previous state-of-the-art in RefCOCO, RefCOCO+ and RefCOCOg.