Large Language Models (LLMs) have achieved remarkable progress in natural language understanding, reasoning, and autonomous decision-making. However, these advancements have also come with significant privacy concerns. While significant research has focused on mitigating the data privacy risks of LLMs during various stages of model training, less attention has been paid to new threats emerging from their deployment. The integration of LLMs into widely used applications and the weaponization of their autonomous abilities have created new privacy vulnerabilities. These vulnerabilities provide opportunities for both inadvertent data leakage and malicious exfiltration from LLM-powered systems. Additionally, adversaries can exploit these systems to launch sophisticated, large-scale privacy attacks, threatening not only individual privacy but also financial security and societal trust. In this paper, we systematically examine these emerging privacy risks of LLMs. We also discuss potential mitigation strategies and call for the research community to broaden its focus beyond data privacy risks, developing new defenses to address the evolving threats posed by increasingly powerful LLMs and LLM-powered systems.