In this paper we present an analysis of the information structural properties of different types of verb and sentence modifying adjuncts under a QUD (question under discussion) approach. Our study is based on naturalistic data from English, French and German containing adjuncts such as temporal, spatial, or manner prepositional phrases, as well as different types of adverbial clauses. The analysis relies on the approach by Riester et al. ( 2018), which identifies the (generally implicit, sometimes overt) QUD preceding each utterance of a text by means of pragmatic principles, and derives from it the information structure of the utterance. The analysis of adjuncts within this approach shows that in certain contexts, despite conveying new information, adjuncts do not answer the QUD that is answered by the sentence they syntactically depend on. We argue that these adjuncts answer a different QUD and behave as independent discourse units. As such, they have an information structure of their own and are in a rhetorical relation with their host clause. Our analysis sheds light on the similarities between adjuncts and Potts' (2005) supplements. Both can be accounted for as independent discourse units; however, while supplements display projective behavior, adjuncts do not. Following Venhuizen et al. (2014), we ascribe this difference to their different semantic anchor (nominal vs. verbal). Our work therefore highlights a different way for an expression to be independent at a discourse level, other than being projective content.