In this modern age of science and technology, the numerical methods such as Boundary Element Methods (BEMs) versus empirical methods have received great attention from researchers and have become more important for the numerical solutions of a number of physical problems in the fields of applied mathematics, physics and engineering. Boundary element method is a numerical technique in which the boundary of body under consideration is subdivided into a series of discrete elements over which the function can vary. The astonishing advances in this method have made it a versatile and powerful technique of computational methods. The method is providing a fertile research area and the field of its applications is continuously widening day by day. This method is superior to the domain type methods such as Finite Difference Method (FDM) and Finite Element Method (FEM), etc. due to its remarkable features. One of the most significant features is the much smaller size of the system of equations and considerable reduction in data, which is pre-requisite to run a computer program efficiently. Moreover, the method is ideally suited to the problems with infinite domains. Therefore, such method is computationally more efficient, accurate, time saving and economical. Boundary element methods can be usually formulated using two different approaches known as the ‘direct’ and ‘indirect’ methods. The direct method takes the form of a statement which provides the values of unknown variables at any field point in terms of the complete set of all the boundary data, whereas the indirect method uses the distribution of singularities over the body surface or the flow field and computes such distribution as the solution for an integral equation. Furthermore, this method is an active area of research in computational fluid dynamics (CFD) and it has been very useful in dealing with fluid flow problems. In this thesis, the author has used different formulations of BEM such as ‘direct’ and ‘indirect’ methods for calculating the solutions for incompressible fluid flow problems. These methods have been implemented on computer using FORTRAN 77. In chapter 0, the basic concepts necessary in the study of Fluid Mechanics are given. In chapter 1, statement of the problem, literature review and the method of solutions are given. The general equations for viscous fluid flow are presented in chapter 2. In chapter 3, equations for boundary element methods are derived. Chapter 4 deals with the discretisation of equation for boundary element method. In chapter 5, the indirect boundary element method has been used to calculate the flow field around two – and three – dimensional bodies. The direct and indirect boundary element methods have been applied to calculate viscous incompressible flow (Oseen flow) around a circular cylinder in chapter 6. Finally, in chapter 7 both the direct and indirect boundary element methods have been used to calculate three – dimensional highly viscous incompressible flow (Creeping flow) past a sphere. It has been observed that the computed results in all the above mentioned cases are in good agreement with the analytical results. At the end, the conclusion and extension for further work have been given.