Alluvial deposits occur over a range of superimposed scales, dependent on the scales of associated topographic features (such as ripples, dunes, bars, channels, floodplains, valleys, river systems), the time and spatial extent over which deposition occurred, and the degree of preservation. Understanding and prediction of alluvial deposits is aided by numerical (forward) modelling of depositional forms and processes. The most desirable approach to such numerical modelling is through solution of the fundamental equations of motion for the fluid and the sediment (e.g. conservation of mass, momentum and energy) for all of the scales of deposits. Construction of the equations of motion requires an understanding of the interactions among unsteady, non-uniform, turbulent water flow, the supplied and transported sediment, and the topography of the sediment bed. This understanding is very incomplete.Simplified numerical forward models have been applied to relatively short-term, small-scale processes such as bed degradation and armouring downstream of dams, reservoir sedimentation, and sorting of sediment (e.g. downstream fining) during deposition in spatially decelerating flows. However, such models are rarely applied over long time periods, over large spatial scales, and where there are complicated temporal and spatial variations in the geometry, water and sediment supply. This is because of limitations to computing facilities, and because of lack of understanding of the workings of the alluvial system. As a result, long-term, large-scale alluvial processes are commonly treated using 'process-imitating' models. Process-imitating models do not necessarily represent processes accurately or completely. This review of process-based models demonstrates that they are generally undeveloped, and linkages between models for different scales are lacking. Process-based modelling is in its infancy. As a result, structure-imitating stochastic models are widely used for simulating alluvial hydrocarbon reservoirs and aquifers, given some initial data. However, such models cannot help understanding of alluvial deposits, nor can they predict their nature outside the data region.