Causal inference has been a pivotal challenge across diverse domains such as medicine and economics, demanding a complicated integration of human knowledge, mathematical reasoning, and data mining capabilities. Recent advancements in natural language processing (NLP), particularly with the advent of large language models (LLMs), have introduced promising opportunities for traditional causal inference tasks. This paper reviews recent progress in applying LLMs to causal inference, encompassing various tasks spanning different levels of causation. We summarize the main causal problems and approaches, and present a comparison of their evaluation results in different causal scenarios. Furthermore, we discuss key findings and outline directions for future research, underscoring the potential implications of integrating LLMs in advancing causal inference methodologies.