In glaciated catchments, glacier-generated floods (Jökulhlaups) put human activity at risk with large, sporadic Jökulhlaups accounting for most flood-related fatalities and damage to infrastructure. In studies of Jökulhlaup hydrodynamics the view predominates that floodwater travels within a distinct conduit eroded into the underside of a glacier. However, some Jökulhlaups produce subglacial responses wholly inconsistent with the conventional theory of drainage. By focusing on Icelandic Jökulhlaups this article reassesses how floodwater flows through glaciers. It is argued that two physically separable classes of Jökulhlaup exist and that not all Jökulhlaups are an upward extrapolation of processes inherent in events of lesser magnitude and smaller scale. The hydraulic coupling of multiple, nonlinear components to the flood circuit of a glacier can induce extreme responses, including pressure impulses in subglacial drainage. Representing such complexity in mathematical form should be the basis for upcoming research, as future modeling results may help to determine the glaciological processes behind Heinrich events. Moreover, such an approach would lead to more accurate, predictive models of Jökulhlaup timing and intensity.