Argument mining targets structures in natural language related to interpretation and persuasion which are central to scientific communication. Most scholarly discourse involves interpreting experimental evidence and attempting to persuade other scientists to adopt the same conclusions. While various argument mining studies have addressed student essays and news articles, those that target scientific discourse are still scarce. This paper surveys existing work in argument mining of scholarly discourse, and provides an overview of current models, data, tasks, and applications. We identify a number of key challenges confronting argument mining in the scientific domain, and suggest some possible solutions and future directions.