This dissertation develops three novel approaches to conceptualize and quantify aspects ofpolitical extremism. My main chapters, which are three standalone papers, specifically examine voters' movement toward more extreme parties and issue positions. A majority of the dissertation focuses on measuring far right extremism due to its prevalence in contemporary democracies. However, Chapter 4 also measures far left radicalization.More broadly, the dissertation is motivated by the discrepancy between publicly expressedand privately held extreme political views. This gap is overlooked in the study of democracies, although it has important implications for measuring and understanding the diffusion of extreme beliefs in democratic electorates.The first chapter (Chapter 2) addresses this gap between public and private preferencesdirectly, through the original concept of contingent extremism. I posit that there are twobroad categories of far right extremists. The first, who I call staunch loyalists, will castfar right votes whenever they are given the opportunity to so. The second, who I callcontingent extremists, will only cast a vote when they believe enough other citizens arealready doing so. Their support is contingent on perceived party popularity. To empiricallytest this concept, I eld survey experiments in Germany (n=1,991), France (n=1,770) andHungary (n=1,015), and measure respondents' willingness to identify as far right supporterswhen randomly assigned to more or less `favorable' polling information about the party.Contingent extremism is captured through the difference in rates of far right identificationin these treatments. To examine geographic variation in contingent extremism, I then matchrespondents to their electoral districts, and find that contingent extremists live in districtswhere the far right has weak electoral support. I use this to derive a local party performancethreshold|roughly one-fifth of vote share|at which they begin to support the party openly.Next, I turn to measuring the gap between the publicly and privately stated positions of farright parties. I posit that far right parties are incentivized to present a more mainstreamideological profile in public communications than they do in private ones. Until recently,information about their private campaign appeals was difficult to obtain, so scholars couldnot address this gap. In the last year, social media data transparency initiatives have madeit possible to explore the microtargeted appeals parties use to mobilize voters online. Thesecond chapter (Chapter 3) details why this new data source is so critical to advancing comparativists' knowledge of far right political strategy. I demonstrate a range of computational tools that can be used to parse the content of ads data, including an unsupervised scaling model of document positions, structural topic modeling, and sentiment analysis. Using these methods, I evaluate the content of more than 68,000 political campaign ads across 11 European countries and 79 political parties, and provide novel insight into the private political campaigns of the far right.Next, I turn to a chapter on measuring polarization, since polarization plays an importantrole in the radicalization of democratic electorates. This third chapter (Chapter 4) detailsa supervised machine learning approach to detect polarization in social media discourse following high salience news events. I operationalize polarization as an increase in the shareof extreme political discourse within a partisan network. My approach is a significant deviation from the current literature, which relies on social network analysis to model network structures and draw inferences about polarization. I argue that a machine learning approach is a necessary supplement, because it evaluates what people actually say about politics. I find that everyday discourse on social media is moderately diverse, but following major news events, political discourse becomes more extreme and partisan. The chapter demonstrates the potential of a supervised learning approach to better understand who is susceptible to polarization and when. Moreover, it offers a path forward for comparative studies of polarization.Finally, Chapter 5 discusses the implications of my findings for extremist politics in democracies and offers future avenues for research. I conclude that far right political views are more common in the electorate than most experts would suggest, in part due to the gapbetween the public and private expression of extreme viewpoints.