The problem of understanding the stream of messages exchanged on social media such as Facebook and Twitter is becoming a major challenge for automated systems. The tremendous amount of data exchanged on these platforms as well as the specific form of language adopted by social media users constitute a new challenging context for existing argument mining techniques. In this paper, we describe an ongoing work towards the creation of a complete argument mining pipeline over Twitter messages: (i) we identify which tweets can be considered as arguments and which cannot, (ii) over the set of tweet-arguments, we group them by topic, and (iii) we predict whether such tweets support or attack each other. The final goal is to compute the set of tweets which are widely recognized as accepted, and the different (possibly conflicting) viewpoints that emerge on a topic, given a stream of messages.