The task of fact-checking deals with assessing the veracity of factual claims based on credible evidence and background knowledge. In particular, scientific fact-checking is the variation of the task concerned with verifying claims rooted in scientific knowledge. This task has received significant attention due to the growing importance of scientific and health discussions on online platforms. Automated scientific fact-checking methods based on NLP can help combat the spread of misinformation, assist researchers in knowledge discovery, and help individuals understand new scientific breakthroughs. In this paper, we present a comprehensive survey of existing research in this emerging field and its related tasks. We provide a task description, discuss the construction process of existing datasets, and analyze proposed models and approaches. Based on our findings, we identify intriguing challenges and outline potential future directions to advance the field.