Detecting anomalies or out-of-distribution (OOD) samples is critical for maintaining the reliability and trustworthiness of machine learning systems. Recently, Large Language Models (LLMs) have demonstrated their effectiveness not only in natural language processing but also in broader applications due to their advanced comprehension and generative capabilities. The integration of LLMs into anomaly and OOD detection marks a significant shift from the traditional paradigm in the field. This survey focuses on the problem of anomaly and OOD detection under the context of LLMs. We propose a new taxonomy to categorize existing approaches into three classes based on the role played by LLMs. Following our proposed taxonomy, we further discuss the related work under each of the categories and finally discuss potential challenges and directions for future research in this field. We also provide an up-to-date reading list of relevant papers.