A flow driven through a finite-length flexible tube or channel can exhibit a rich array of instabilities.  As well as having relevance to blood vessels and lung airways, the mechanisms of instability have been a long-standing conundrum in physiological fluid mechanics.  I will describe recent progress towards understanding their origin revealed by a variety of modelling approaches.  A spatially one-dimensional model of flexible-channel flow illustrates the relationship between local modes and a global “sloshing” mode.  A more accurate two-dimensional model shows how a local flutter mode originates through a novel “weak” critical layer, while also contributing to a global instability.  I will also explain how asymptotic methods can be used to predict stability thresholds in a three-dimensional collapsible tube.  This is joint work with Jonathan Boyle, Matthias Heil, Peter Stewart, Sarah Waters and Robert Whittaker.