Ensuring and improving the safety of autonomous driving systems (ADS) is crucial for the deployment of highly automated vehicles, especially in safety-critical events. To address the rarity issue, adversarial scenario generation methods are developed, in which behaviors of traffic participants are manipulated to induce safety-critical events. However, existing methods still face two limitations. First, identification of the adversarial participant directly impacts the effectiveness of the generation. However, the complexity of real-world scenarios, with numerous participants and diverse behaviors, makes identification challenging. Second, the potential of generated safety-critical scenarios to continuously improve ADS performance remains underexplored. To address these issues, we propose LLM-attacker: a closed-loop adversarial scenario generation framework leveraging large language models (LLMs). Specifically, multiple LLM agents are designed and coordinated to identify optimal attackers. Then, the trajectories of the attackers are optimized to generate adversarial scenarios. These scenarios are iteratively refined based on the performance of ADS, forming a feedback loop to improve ADS. Experimental results show that LLM-attacker can create more dangerous scenarios than other methods, and the ADS trained with it achieves a collision rate half that of training with normal scenarios. This indicates the ability of LLM-attacker to test and enhance the safety and robustness of ADS. Video demonstrations are provided at: https://drive.google.com/file/d/1Zv4V3iG7825oyiKbUwS2Y-rR0DQIE1ZA/view.