We provide a selective review of recent progress in the analysis of several physiological and physiologically-inspired fluid-structure interaction problems; our aim being to explain the underlying physical mechanisms that cause the observed behaviours. Specifically, we discuss recent studies of self-excited oscillations in collapsible tubes, focusing primarily on studies of an idealised model system: the Starling resistor --- a device used in most laboratory experiments. We next review studies of a particular physiological, flow-induced oscillation: vocal-fold oscillations during phonation. Finally, we discuss the closure and reopening of pulmonary airways, physiological fluid-structure interaction problems that also involve the airways' liquid lining.