Mathematical modelling provides an essential tool for understanding aeolian transport of dust and sand, its spatial and temporal dynamics, and its role in the earth system. As much of the crucial action takes place at or just above land surfaces, adequate representation of land-surface processes is essential. In this paper, we review progress in aeolian transport modelling during last two decades, with emphasis on the representation of several key land-surface processes: (1) saltation; (2) dust uplift; (3) the threshold for particle motion, including the effects of surface sheltering and cohesion; (4) turbulent diffusion near the ground; and (5) deposition of particles to the surface. These process models are then placed in the context of integrated models of aeolian transport, in which the conservation equation for atmospheric dust is solved over a large region or the globe. We identify and assess four challenges in large-scale aeolian transport modelling: (1) the fidelity of process representations, (2) upscaling point-scale process models in the presence of unresolved heterogeneity in space and time (where a new approach is suggested), (3) availability of spatial data on model inputs and boundary conditions, and (4) large-scale parameter estimation.