In the last decade, the field of argument mining has grown notably. However, only relatively few studies have investigated argumentation in social media and specifically on Twitter. Here, we provide the, to our knowledge, first critical in-depth survey of the state of the art in tweet-based argument mining. We discuss approaches to modelling the structure of arguments in the context of tweet corpus annotation, and we review current progress in the task of detecting argument components and their relations in tweets. We also survey the intersection of argument mining and stance detection, before we conclude with an outlook.

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