Linguistic analyses of the Question Under Discussion (QUD) provide an interesting extension to Tony Sanford's work on discourse coherence (e.g., Sanford & Emmott, 2012). The QUD approach claims that discourse is organized by a series of overt and covert questions and answers to, or comments on, them. In a coherent discourse, material that addresses the current QUD receives focus, and its processing is facilitated. After a brief review of the existing literature showing that the QUD affects discourse processing, we describe several lines of our own published research on the QUD and briefly present two new experiments. We first review the effects of overt questions on sentence comprehension, arguing that sentences that are thematically and syntactically congruent with the bias of an overt preceding question are processed faster than ones that require a syntactic and thematic shift. We present new evidence suggesting that assertions that address the focused element of a preceding question are likely to receive an exhaustive interpretation. Turning to covert questions, we review evidence that modals and other indications of the uncertain possibility of some event introduce a QUD about the occurrence of that event, biasing and facilitating comprehension of sentences that address such a QUD. Finally, we present new evidence suggesting that although sentential assertions in a discourse are likely to be taken to address an implicit QUD introduced by the antecedent of a conditional, elliptical assertions still tend to find their antecedents in a phrase whose structure is readily available. Thus, although satisfying the requirements of the current QUD is one factor that affects discourse comprehension, it is actually just one of many.