With more than 3 billion users, online social networks represent an important venue for moral and political discourse and have been used to organize political revolutions, influence elections, and raise awareness of social issues. These examples rely on a common process to be effective: the ability to engage users and spread moralized content through online networks. Here, we review evidence that expressions of moral emotion play an important role in the spread of moralized content (a phenomenon we call moral contagion). Next, we propose a psychological model called the motivation, attention, and design (MAD) model to explain moral contagion. The MAD model posits that people have group-identity-based motivations to share moral-emotional content, that such content is especially likely to capture our attention, and that the design of social-media platforms amplifies our natural motivational and cognitive tendencies to spread such content. We review each component of the model (as well as interactions between components) and raise several novel, testable hypotheses that can spark progress on the scientific investigation of civic engagement and activism, political polarization, propaganda and disinformation, and other moralized behaviors in the digital age.