This dissertation investigates the syntactic and semantic properties of fragments -- utterances which consist of a constituent smaller than a clause. Examples include short answers, such as What did he eat? --- Chips, as well as cases which do not respond to any overt question; for example, saying The train station, please on entering a taxi. I defend Merchant 2004's proposal that, underlyingly, fragments contain clausal structure: the fragment answer chips is elliptical for he ate chips, with he ate being present in the syntax but unspoken. I argue that challenges to ellipsis-based accounts of fragments can be circumvented by adopting a particular semantic restriction on which clauses are allowed to elide. Building on an analysis by Reich 2007, I argue that elided clauses must stand in a particular relation to Roberts 2012/1996's Question under Discussion, which I dub QUD-GIVENness I also discuss the syntactic properties of fragments. Merchant 2004 argues that fragments are generated by A'-movement to the left periphery. However, I show that by other diagnostics, fragments appear not to have moved. I solve this contradiction by arguing that fragments do move, but that this movement takes place only at the level of Phonological Form. At Logical Form, the fragment remains in situ. It is this 'split' which causes some diagnostics for movement to succeed and others to fail. Finally, the dissertation considers cases of embedded fragments, such as Who ate the cookies? --- I think John. Fragments can only be embedded in this way under bridge verbs. Following many authors, I assume that bridge verbs embed a double-complementizer or 'recursive CP' structure, while other clausal-embedding verbs embed clauses with only one complementizer. I argue that the 'higher' complementizer head embedded by bridge verbs is the head which licenses clausal ellipsis. I support this hypothesis by investigating which wh-movement structures allow sluicing. I argue that the wh-movement structures which allow sluicing are just those which can be argued to have a double complementizer/recursive CP structure, providing evidence for the hypothesis that the 'higher' complementizer in these structures is the licensor of clausal ellipsis.