Political debates offer a rare opportunity for citizens to compare the candidates’ positions on the most controversial topics of the campaign. Thus they represent a natural application scenario for Argument Mining. As existing research lacks solid empirical investigation of the typology of argument components in political debates, we fill this gap by proposing an Argument Mining approach to political debates. We address this task in an empirical manner by annotating 39 political debates from the last 50 years of US presidential campaigns, creating a new corpus of 29k argument components, labeled as premises and claims. We then propose two tasks: (1) identifying the argumentative components in such debates, and (2) classifying them as premises and claims. We show that feature-rich SVM learners and Neural Network architectures outperform standard baselines in Argument Mining over such complex data. We release the new corpus USElecDeb60To16 and the accompanying software under free licenses to the research community.