What makes a discourse coherent? One potential factor has been discussed in the linguistic literature in terms of a Question under Discussion (QUD). This approach claims that discourse proceeds by continually raising explicit or implicit questions, viewed as sets of alternatives, or competing descriptions of the world. If the interlocutor accepts the question, it becomes the QUD, a narrowed set of alternatives to be addressed (Roberts, in press). Three eye movement recording studies are reported that investigated the effect of a preceding explicit QUD (Experiment 1) or implicit QUD (Experiments 2 and 3) on the processing of following text. Experiment 1 revealed an effect of whether the question queried alternative propositions or alternative entities. Reading times in the answer were faster when the answer it provided was of the same semantic type as was queried. Experiment 2 tested QUDs implied by the alternative description of reality introduced by a non-actuality implicature trigger such as "should X" or "want to X". The results, when combined with the results of Experiment 3 (which ruled out a possible alternative interpretation) showed disrupted reading of a following verb phrase that failed to resolve the implicit QUD ("Did the discourse participant actually X?"), compared to reading the same material in the absence of a clear QUD. The findings support an online role for QUDs in guiding readers' structuring and interpretation of discourse. (Contains 6 tables and 2 figures.)