The phenomenon of empathy entails the ability to share the affective experiences of others. In recent years social neuroscience made considerable progress in revealing the mechanisms that enable a person to feel what another is feeling. The present review provides an in-depth and critical discussion of these findings. Consistent evidence shows that sharing the emotions of others is associated with activation in neural structures that are also active during the first-hand experience of that emotion. Part of the neural activation shared between self- and other-related experiences seems to be rather automatically activated. However, recent studies also show that empathy is a highly flexible phenomenon, and that vicarious responses are malleable with respect to a number of factors—such as contextual appraisal, the interpersonal relationship between empathizer and other, or the perspective adopted during observation of the other. Future investigations are needed to provide more detailed insights into these factors and their neural underpinnings. Questions such as whether individual differences in empathy can be explained by stable personality traits, whether we can train ourselves to be more empathic, and how empathy relates to prosocial behavior are of utmost relevance for both science and society.