Human language offers rich ways to track, compare, and engage the attentional and epistemic states of interlocutors. While this task is central to everyday communication, our knowledge of the cross-linguistic grammatical means that target such intersubjective coordination has remained basic. In two serialised papers, we introduce the term â€˜engagementâ€™ to refer to grammaticalised means for encoding the relative mental directedness of speaker and addressee towards an entity or state of affairs, and describe examples of engagement systems from around the world. Engagement systems express the speakerâ€™s assumptions about the degree to which their attention or knowledge is shared (or not shared) by the addressee. Engagement categories can operate at the level of entities in the here-and-now (deixis), in the unfolding discourse (definiteness vs indefiniteness), entire event-depicting propositions (through markers with clausal scope), and even metapropositions (potentially scoping over evidential values). In this first paper, we introduce engagement and situate it with respect to existing work on intersubjectivity in language. We then explore the key role of deixis in coordinating attention and expressing engagement, moving through increasingly intercognitive deictic systems from those that focus on the the location of the speaker, to those that encode the attentional state of the addressee.