Incidental adjuncts are adjuncts which are intonationally set apart from the rest of the sentence (e.g. 'John was here, it seems.'). Incidentality is a phonological/syntactic property, distinct from the semantic/pragmatic property of 'parentheticality'. We argue that at least in French, the distribution and scopal properties are incidentals are at odds with those of other adjuncts, and thus present a challenge for current approaches to the grammar of adjuncts. We propose an analysis of French incidental modifiers which integrates seamlessly with current accounts of nonincidental modifiers and requires only conservative extensions to phrase structure grammar, along the lines of Reape's (1994) work on order domains.