In a surprising turn, Large Language Models (LLMs) together with a growing arsenal of prompt-based heuristics now offer powerful off-the-shelf approaches providing few-shot solutions to myriad classic NLP problems. However, despite promising early results, these LLM-based few-shot methods remain far from the state of the art in Named Entity Recognition (NER), where prevailing methods include learning representations via end-to-end structural understanding and fine-tuning on standard labeled corpora. In this paper, we introduce PromptNER, a new state-of-the-art algorithm for few-Shot and cross-domain NER. To adapt to any new NER task PromptNER requires a set of entity definitions in addition to the standard few-shot examples. Given a sentence, PromptNER prompts an LLM to produce a list of potential entities along with corresponding explanations justifying their compatibility with the provided entity type definitions. Remarkably, PromptNER achieves state-of-the-art performance on few-shot NER, achieving an 11% (absolute) improvement in F1 score on the ConLL dataset, and a 10% (absolute) improvement on the FewNERD dataset. PromptNER also moves the state of the art on Cross Domain NER, outperforming all prior methods (including those not limited to the few-shot setting), setting a new mark on all 5 CrossNER target domains, with an average F1 gain of 9%, despite using less than 2% of the available data.