This thesis is concerned with the numerical solution of the steady state wave equation - Helmholtz equation - exterior to one or several bodies positioned in free space. The current work may without complications be applied to interior problems and problems concerning other fluids in which Helmholtz equation is valid. The approximate numerical solution to an acoustic radiation or scattering problem is obtained by bringing Helmholtz equation to its integral form: Helmholtz integral equation. Helmholtz integral equation is then solved numerically by means of the Boundary Element Method (BEM). The boundary element method is suitable for the approximate numerical solution of exterior acoustic problems due to two features: i) the radiation condition is automatically satisfied, and ii) only the boundary of the domain in interest needs to be discretized. During the course of this study computer programs have been developed for calculating the sound field exterior to bodies of axisymmetric or general three-dimensional shape, positioned in free space. Since this is the first study of the boundary element method in acoustics carried out at the Acoustics Laboratory, the emphasis has been on general aspects of the method rather than on details. The author hopes that this thesis may serve as a basis for further investigations of the boundary element method.