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The intersection of social media and politics is yet another realm in which Computational Social Science has a paramount role to play. In this review, I examine the questions that computational social scientists are attempting to answer – as well as the tools and methods they are developing to do so – in three areas where the rise of social media has led to concerns about the quality of democracy in the digital information era: online hate; misinformation; and foreign influence campaigns. I begin, however, by considering a precursor of these topics – and also a potential hope for social media to be able to positively impact the quality of democracy – by exploring attempts to measure public opinion online using Computational Social Science methods. In all four areas, computational social scientists have made great strides in providing information to policy makers and the public regarding the evolution of these very complex phenomena but in all cases could do more to inform public policy with better access to the necessary data; this point is discussed in more detail in the conclusion of the review.