We describe how surface-tension-driven instabilities of the lung's liquid lining may lead to pulmonary airway closure via the formation of liquid bridges that occlude the airway lumen. Using simple theoretical models, we demonstrate that this process may occur via a purely fluid-mechanical "film collapse" or through a coupled, fluid-elastic "compliant collapse" mechanism. Both mechanisms can lead to airway closure in times comparable with the breathing cycle, suggesting that surface tension is the primary mechanical effect responsible for the closure observed in peripheral regions of the human lungs. We conclude by discussing the influence of additional effects not included in the simple models, such as gravity, the presence of pulmonary surfactant, respiratory flow and wall motion, the airways' geometry, and the mechanical structure of the airway walls.