Engagement is a concept of the utmost importance in human-computer interaction, not only for informing the design and implementation of interfaces, but also for enabling more sophisticated interfaces capable of adapting to users. While the notion of engagement is actively being studied in a diverse set of domains, the term has been used to refer to a number of related, but different concepts. In fact it has been referred to across different disciplines under different names and with different connotations in mind. Therefore, it can be quite difficult to understand what the meaning of engagement is and how one study relates to another one accordingly. Engagement has been studied not only in human-human, but also in human-agent interactions i.e., interactions with physical robots and embodied virtual agents. In this overview article we focus on different factors involved in engagement studies, distinguishing especially between those studies that address task and social engagement, involve children and adults, are conducted in a lab or aimed for long term interaction. We also present models for detecting engagement and for generating multimodal behaviors to show engagement.