The movement of cohesive sediment is of great importance in many coastal and estuarine engineering problems. Navigation channels often used to be dredged to maintain navigable depths, allowing for the effect of a harbour or wharf on the local sediment transport regime. Contaminants are readily absorbed by silt and clay particles, causing a range of water quality problems. This thesis describes the development and testing of a finite element program to model cohesive sediment transport. The program solves the coupled Navier-Stokes and scalar-transport equations along with several complex numerical algorithms for settling velocity, flocculation, non-Newtonian flow and turbulence. The program also uses h-adaptivity and unstructured mesh generation to capture important flow features. The program is benchmarked against the thermally driven cavity problem, producing results that compare well with existing solutions without any special scheme for advection dominated flow. This is possible by modelling the transient problem using h-adaptivity. The programme is also applied to realistic cohesive sediment transport problems. It predicts the formation of a hindered settling layer and uses h-adaptivity to capture sharp density interfaces. It also solves settling of dredged material onto a inclined bed and non-Newtonian flow in a race-track flume. The program produces results that compare well with experimental data. The h-adaptive finite element method is found to be a very successful in modelling the transport of cohesive sediment and its associated physical processes.