In this thesis, a semi-analytical and meshless method is proposed to solve free-surface fluid problems. Two approaches, one based on using the pressure field as the potentaial of the accelerations and another based on using a velocity potential, areemplpoyedto solve the equations in 3-dimension. In the first approach, Navier-Stoks equations for incompressible inviscidfluids convert into a pressure Laplace equation which is solved, in each time step, by a meshless method using residual-free basis functions. Subsequently, fluid parameters such as accelerations, velocites and displacements are calculated. The boundary conditions of the problem are defined in terms of acclerations (at the walls) and the pressure (at the free surface).The free surface of the fluid is updated using a Lagrangian approach. The solution to the pressure field is approximatedby as a series of exponential basis functions (EBFs). These functions are selected in a manner so that they exactly satisfy the pressure Laplace equation. The coefficients of the series are determined by satisfying the boundary conditions over a selection of boundary points using a collocation scheme through a special discrete transformation. In the second approach, i.e. using velocity potential for incompressible inviscid fluids, Navier-Stokes equations convertinto another Laplace equation (i.e. for the potential function). The boundary conditions of the problem in this case are thus d efined in terms of velocities (at the walls) and the value of the potential function (at the free surface). Using the meshlessfor this latter problem, velocities and thus displacementsare determined. Since in both aforementioned approaches, the solution accuracy of the3D Laplace equation affects the convergence of the solution over time, a new algorithm for the selection of the EBFs is introduced in this thesis. Linear and non-linear sloshings in cubic and cylinderical tanks are considered for the validation of the resultswhich are compared with those of other Lagrangian methods. Significantly lower computational time is one of the advantages of the proposed method. Key words : Exponential basis functions, Meshless method, Navier-Stokes equations, Lagrangian free surface flow, Pressure theory, Ptential flow theory.