Existing Medical Large Vision-Language Models (Med-LVLMs), encapsulating extensive medical knowledge, demonstrate excellent capabilities in understanding medical images. However, there remain challenges in visual localization in medical images, which is crucial for abnormality detection and interpretation. To address these issues, we propose a novel UMed-LVLM designed to unveil medical abnormalities. Specifically, we collect a Medical Abnormalities Unveiling (MAU) dataset and propose a two-stage training method for UMed-LVLM training. To collect MAU dataset, we propose a prompt method utilizing the GPT-4V to generate diagnoses based on identified abnormal areas in medical images. Moreover, the two-stage training method includes Abnormal-Aware Instruction Tuning and Abnormal-Aware Rewarding, comprising Relevance Reward, Abnormal Localization Reward and Vision Relevance Reward. Experimental results demonstrate that our UMed-LVLM significantly outperforms existing Med-LVLMs in identifying and understanding medical abnormalities, achieving a 58% improvement over the baseline. In addition, this work shows that enhancing the abnormality detection capabilities of Med-LVLMs significantly improves their understanding of medical images and generalization capability.