- This paper provides an introduction to a collection of papers focusing on the left and right peripheries in discourse from both theoretical and empirical perspectives as well as across various languages. After outlining the main aims and questions underlying research on the left and right peripheries of discourse units, this paper presents an overview of the current state of the art by summarizing some of the key topics that have recently received increasing scholarly attention. These include (i) the question of granularity, i.e. the nature of the unit in relation to which the left and right peripheries can be defined, (ii) the question of how the left and right peripheries are internally structured and linguistically realized, and (iii) the question of where to draw the boundary between the core and the peripheries of discourse units. This introduction concludes with short summaries of the six papers included in this special issue.