Large language models (LLMs) have recently shown remarkable performance in language tasks and beyond. However, due to their limited inherent causal reasoning ability, LLMs still face challenges in handling tasks that require robust causal reasoning ability, such as health-care and economic analysis. As a result, a growing body of research has focused on enhancing the causal reasoning ability of LLMs. Despite the booming research, there lacks a survey to well review the challenges, progress and future directions in this area. To bridge this significant gap, we systematically review literature on how to strengthen LLMs' causal reasoning ability in this paper. We start from the introduction of background and motivations of this topic, followed by the summarisation of key challenges in this area. Thereafter, we propose a novel taxonomy to systematically categorise existing methods, together with detailed comparisons within and between classes of methods. Furthermore, we summarise existing benchmarks and evaluation metrics for assessing LLMs' causal reasoning ability. Finally, we outline future research directions for this emerging field, offering insights and inspiration to researchers and practitioners in the area.