We focus on the problem of sediment transport by the action of some fluid such as an ocean or a river. Sediment transport is usually classified in two main modes: bed load and suspended load. Bed load transport is usually modeled by the Exner system. This model may be completed and generalized including suspended sediment transport and other more complex effects. We propose here to review classical models for sediment transport in shallow water as well as recent advances in this field. The aim is to better describe the physics of the problem. Numerical simulations will be done under the general framework of path-conservative numerical schemes.