This study presents a new Lagrangian model for predicting dynamics and transport in rivers and shallow water flows. A hydrostatic model is developed for the prediction of rivers and floodplain flow and lateral interactions between them. The model is extended to the Boussinesq weakly non-linear, non-hydrostatic model for the simulation of solitary waves and undular bores. A model for advection-diffusion transport of tracers in open channel flow is also presented. The simulation results are compared against an analytical solution and published laboratory data, field data and theoretical results. It is demonstrated that the Lagrangian moving grid eliminates numerical diffusion and oscillations; the model is dynamically adaptive, providing higher resolution under the wave by compressing the parcels (grid). It also allows flow over dry beds and moving boundaries to be handled efficiently. The hydrostatic model results have shown that the model accurately simulates wave propagation and non-linear steepening until wave breaking. The model is successfully applied to simulate flow and lateral interactions in a compound channel and flood wave movement in a natural river. The non-hydrostatic model has successfully reproduced the general features of solitary waves such as the balance between non-linearity and wave dispersion and non-linear interactions of two solitary waves by phase-shift. Also, the model successfully reproduced undular bores (high frequency short waves) from a long wave and the predicted maximum height of the leading wave agreed very well with the published results. It is shown that the simple second order accurate Lagrangian scheme efficiently simulates dispersive waves without any numerical diffusion. Lagrangian modeling of advection-diffusion transport of Gaussian tracer distributions, top hat tracer distributions and steep fronts (step function) in steady, uniform flow has provided exact results and has shown that the scheme allows the use of a large time step without any numerical diffusion and oscillations, including for the advection of steep fronts. The scheme can handle large Courant numbers (results are presented for Cr = 0 to 20) and the entire range of grid Peclet numbers from zero to infinity. The model is successfully applied to tracer transport due to flow induced by simple waves, solitary waves and undular bores