The advancement of large language models (LLMs) has significantly enhanced the ability to effectively tackle various downstream NLP tasks and unify these tasks into generative pipelines. On the one hand, powerful language models, trained on massive textual data, have brought unparalleled accessibility and usability for both models and users. On the other hand, unrestricted access to these models can also introduce potential malicious and unintentional privacy risks. Despite ongoing efforts to address the safety and privacy concerns associated with LLMs, the problem remains unresolved. In this paper, we provide a comprehensive analysis of the current privacy attacks targeting LLMs and categorize them according to the adversary's assumed capabilities to shed light on the potential vulnerabilities present in LLMs. Then, we present a detailed overview of prominent defense strategies that have been developed to counter these privacy attacks. Beyond existing works, we identify upcoming privacy concerns as LLMs evolve. Lastly, we point out several potential avenues for future exploration.