The rapid spread of radical ideologies in recent years has led to a worldwide string of terrorist attacks. Understanding how extremist tendencies germinate, develop, and drive individuals to action is important from a cultural standpoint, but also to help formulate response and prevention strategies. Demographic studies, interviews with radicalized subjects, analysis of terrorist databases, reveal that the path to radicalization occurs along progressive steps, where age, social context and peer-to-peer exchange of extremist ideas play major roles. Furthermore, the advent of social media has offered new channels of communication, facilitated recruitment, and hastened the leap from mild discontent to unbridled fanaticism. While a complete sociological understanding of the processes and circumstances that lead to full-fledged extremism is still lacking, quantitative approaches, using modeling and data analyses, can offer useful insight. We review some approaches from statistical mechanics, applied mathematics, data science, that can help describe and understand radicalization and terrorist activity. Specifically, we focus on compartment models of populations harboring extremist views, continuous time models for age-structured radical populations, radicalization as social contagion processes on lattices and social networks, adversarial evolutionary games coupling terrorists and counter-terrorism agents, and point processes to study the spatiotemporal clustering of terrorist events. We also present recent applications of machine learning methods on open-source terrorism databases. Finally, we discuss the role of institutional intervention and the stages at which de-radicalization strategies might be most effective.