The study of influence-driven propagations in social networks and its exploitation for viral marketing purposes has recently received a large deal of attention. However, regardless of the fact that users' authoritativeness, expertise, trust and influence are evidently topic-dependent, the research on social influence has surprisingly largely overlooked this aspect. In this article, we study social influence from a topic modeling perspective. We introduce novel topic-aware influence-driven propagation models that, as we show in our experiments, are more accurate in describing real-world cascades than the standard (i.e., topic-blind) propagation models studied in the literature. In particular, we first propose simple topic-aware extensions of the well-known Independent Cascade and Linear Threshold models. However, these propagation models have a very large number of parameters which could lead to overfitting. Therefore, we propose a different approach explicitly modeling authoritativeness, influence and relevance under a topic-aware perspective. Instead of considering user-to-user influence, the proposed model focuses on user authoritativeness and interests in a topic, leading to a drastic reduction in the number of parameters of the model. We devise methods to learn the parameters of the models from a data set of past propagations. Our experimentation confirms the high accuracy of the proposed models and learning schemes.