Many fluid-conveying vessels in the human body are elastic and can undergo significant flow-induced deformations, making physiological fluid mechanics a rich source of large-displacement fluid–structure interaction problems. The aim of this chapter is to demonstrate the effects of wall elasticity on three canonical internal flows that arise in physiological applications. First, we discuss low-Reynolds number flows in the Starling resistor—the elastic-walled equivalent of Hagen–Poiseuille flow—as a generic model for single-phase flows in elastic vessels. This requires coupled solution of the Stokes equations and the equations of large-displacement elasticity. Next, we extend the theoretical framework to incorporate the presence of air–liquid interfaces and study the propagation of an air finger into a fluid-filled, non-axisymmetrically collapsed vessel—the fluid–structure interaction equivalent of the “Bretherton problem”, a model of pulmonary airway (re-)opening. Finally, we examine the effect of wall elasticity on the Rayleigh–Plateau instability and show that fluid–structure interaction facilitates the formation of occluding liquid bridges in liquid-lined elastic vessels—a scenario of relevance to the physiological problem of pulmonary airway closure. Throughout this chapter, we focus on the study of idealised model problems whose relative simplicity allows us to identify the primary physical mechanisms that underlie the observed behaviour.