It is proposed to study the Boussinesq equations of water wave theory from modelling, analysis and numerical approximation points of view. These equations consist of systems of nonlinear partial diffrential equations of evolution that model two-way propagation of long waves of small amplitude on the water surface. We will first study the well-posedness of new initial-boundary value problems (ibvp's) of physical interest for these systems in 1 and 2D, modelling surface wave flows over horizontal bottoms. We will construct efficient numerical methods for these systems and prove rigorous stability and convergence estimates. For Boussinesq systems modelling waves over bottoms of variable topography, we will study the well-posedness of associated ibvp's in 1 and 2D and prove error estimates for fully discrete finite element methods for their numerical approximation. Finally, we will develop an efficient finite element computer code for the simulation of solutions of Boussinesq systems with variable bottom, with the aim of using it in tsunami propagation studies. We will equip the code with tsunami source mechanisms and with empirical regridding techniques in one and two space dimensions to simulate tsunami run-up on the coast.