The main goal of the thesis is to solve the inverse acoustic scattering problem. The purpose of the thesis is first to study methods for solving mathematical inverse problems. Secondly, to solve the inverse acoustic scattering problem ; to determine the shape of the obstacle that scatters an incident plane wave when some knowledge about the total field of the scattered wave is given, outside of the obstacle (inverse exterior problem for Helmholtz equation). The Direct Variational method is applied and a computer program, implementing the method is carried out for the solution of an inverse problem involving an elliptic PDE. Numerical simulation for both interior and exterior problems for non-homogeneous and homogeneous Helmholtz equation respectively is utilized. All results are given either graphical and/or tabulated forms, indicating that the Direct Variational method is reliable in solving mathematical inverse problems including the inverse acoustic scattering problem.