In this dissertation, local form of Exponential Basis Functions (EBFs) method is used to solve Laplace equation governing incompressible inviscid free surface fluid flow. Generally, mass conservation and momentum conservation equations, known as Navier-Stokes equations, govern isothermal fluid flows. Assuming incompressible inviscid fluid flow, the aforementioned equations convert into a Laplace equation in terms of the fluid’s pressure when a Lagrangian description is employed. Acceleration of the particles is calculated by solving such a Laplace equation with a set of boundary conditions. Likewise, Navier-Stokes equations convert into Laplace equation for the velocity potential if, again, incompressible inviscid non-rotational fluid flow is assumed. By solving this equation, the velocity of the particles is calculated. Focusing on Laplace equation in either of these cases, in this investigation local form of exponential basis functions is used to solve the problems. In this method, a series of exponential basis functions, each one satisfying Laplace equation, is assumed as an approximate solution for the equation within a cloud (representing a subdomain). The boundary conditions are then satisfied in the clouds constructed at boundaries. Therefore, the method can be categorized in meshless/integral-free methods. This is one of the advantages of the method in solving time-dependent problems, in this dissertation free-surface flows, using a time marching algorithm. To illustrate the capabilities of the method, some free-surface fluid problems are solved and the accuracy of method is demonstrated. Key words : Exponential basis functions, Meshless method, Navier-Stokes equations, Lagrangian fluid flow, Pressure theory, Velocity Potential theory.