Mathematical modelling of the fluid mechanics of mucociliary clearance (MCC) is reviewed and future challenges for researchers are discussed. The morphology of the bronchial and tracheal airway surface liquid (ASL) and ciliated epithelium are briefly introduced. The cilia beat cycle, beat frequency and metachronal coordination are described, along with the rheology of the mucous layer. Theoretical modelling of MCC from the late 1960s onwards is reviewed, and distinctions between 'phenomenological', 'slender body theory' and recent 'fluid-structure interaction' models are explained. The ASL consists of two layers, an overlying mucous layer and underlying watery periciliary layer (PCL) which bathes the cilia. Previous models have predicted very little transport of fluid in the PCL compared with the mucous layer. Fluorescent tracer transport experiments on human airway cultures conducted by Matsui et al. [Matsui, H., Randell, S.H., Peretti, S.W., Davis, C.W., Boucher, R.C., 1998. Coordinated clearance of periciliary liquid and mucus from airway surfaces. J. Clin. Invest. 102 (6), 1125-1131] apparently showed equal transport in both the PCL and mucous layer. Recent attempts to resolve this discrepancy by the present authors are reviewed, along with associated modelling findings. These findings have suggested new insights into the interaction of cilia with mucus due to pressure gradients associated with the flat PCL/mucus interface. This phenomenon complements previously known mechanisms for ciliary propulsion. Modelling results are related to clinical findings, in particular the increased MCC observed in patients with pseudohypoaldosteronism. Recent important advances by several groups in modelling the fluid-structure interaction by which the cilia movement and fluid transport emerge from specification of internal mechanics, viscous and elastic forces are reviewed. Finally, we discuss the limitations of existing work, and the challenges for the next generation of models, which may provide further insight into this complex and vital system.