Large Language Models (LLMs) have achieved remarkable success on question answering (QA) tasks, yet they often encode harmful biases that compromise fairness and trustworthiness. Most existing bias mitigation approaches are restricted to predefined categories, limiting their ability to address novel or context-specific emergent biases. To bridge this gap, we tackle the novel problem of open-set bias detection and mitigation in text-based QA. We introduce OpenBiasBench, a comprehensive benchmark designed to evaluate biases across a wide range of categories and subgroups, encompassing both known and previously unseen biases. Additionally, we propose Open-DeBias, a novel, data-efficient, and parameter-efficient debiasing method that leverages adapter modules to mitigate existing social and stereotypical biases while generalizing to unseen ones. Compared to the state-of-the-art BMBI method, Open-DeBias improves QA accuracy on BBQ dataset by nearly $48\%$ on ambiguous subsets and $6\%$ on disambiguated ones, using adapters fine-tuned on just a small fraction of the training data. Remarkably, the same adapters, in a zero-shot transfer to Korean BBQ, achieve $84\%$ accuracy, demonstrating robust language-agnostic generalization. Through extensive evaluation, we also validate the effectiveness of Open-DeBias across a broad range of NLP tasks, including StereoSet and CrowS-Pairs, highlighting its robustness, multilingual strength, and suitability for general-purpose, open-domain bias mitigation. The project page is available at: https://sites.google.com/view/open-debias25