The pulmonary airway system is a highly hierarchical tree-like 3D network in charge of transferring oxygen from the upper airways down to the alveolar region where the gas exchange occurs between air and blood. It is also the place of numerous particle and fluid transport processes. Given the complexity of the anatomy and of the physics involved, numerical simulation is a remarkable tool for investigating the properties of this organ seen as a "transport system", understanding its behavior and possible failures, and predicting the outcome of therapies. However, the range of scales represents here a huge challenge: from the meter scale of the entire organ size down to the micron scale of the mucus layer, about 5 to 6 orders of magnitude of length scale are crossed.

In this talk, we will present several examples of numerical models able to capture the various aspects of the lung airway system, from the simplest linear approach of gas transport to more advanced computational fluid dynamics simulations. We will show, in particular, how the range of scales involved and the difficulty to access the actual parameters in the patient imposes to use a hierarchy of models and a diversity of numerical techniques. We will explain how these models can be used either to reach a general understanding of the system or to design patient specific diagnosis and therapy.