Unsupervised and zero-shot Named Entity Recognition (NER) aims to extract and classify entities in documents from a target domain without annotated data. This setting is particularly relevant for specific domains (biomedical, legal, scientific, ...) where labeled documents are scarce and expensive to create. While zero-shot NER approaches yield impressive outcomes, they operate under the assumption that all entity types are predefined and known. This limitation makes their application impossible in novelty detection, exploration, or knowledge graph construction scenarios. To address this shortcoming, we introduce OWNER, our unsupervised and open-world NER model, which does not need annotations in the target domain (similar to zeroshot) and does not require knowledge of the target entity types or their number. We propose a novel triangular architecture to type and structure entities automatically. It comprises a prompt-based entity type encoder, an unsupervised clustering model, and embedding refinement with contrastive learning to refine entity embeddings and elicit entity types more precisely. Results on 13 domain-specific datasets show that OWNER outperforms open-world large language model prompting (4 % -18 % in AMI) and performs competitively with state-of-the-art zero-shot models. Qualitative analysis shows that OWNER effectively groups entities into semantically meaningful clusters that closely resemble actual entity types (without knowing them beforehand).