As large language models (LLMs) become increasingly prevalent across many real-world applications, understanding and enhancing their robustness to adversarial attacks is of paramount importance. Existing methods for identifying adversarial prompts tend to focus on specific domains, lack diversity, or require extensive human annotations. To address these limitations, we present Rainbow Teaming, a novel black-box approach for producing a diverse collection of adversarial prompts. Rainbow Teaming casts adversarial prompt generation as a quality-diversity problem and uses open-ended search to generate prompts that are both effective and diverse. Focusing on the safety domain, we use Rainbow Teaming to target various state-of-the-art LLMs, including the Llama 2 and Llama 3 models.