This thesis is divided into four parts; while the first part is an introductory one, the remainder constitutes the main body of the research. Apart from the seventh Chapter and unless otherwise acknowledged, the research carried out is claimed as original. The thesis is devoted to single and parallel machine scheduling problems where the objective is to minimize total weighted completion time. The branch and bound approach is adopted throughout to find an optimal solution. As the major component in designing such methods is the lower bound, the second part of the thesis deals with job splitting techniqe. This is extended and generalized so that it generates a class of bounds for many problems. It is described as a methodology for this purpose. The third part however is concerned with another bounding scheme, known as Lagrangian relaxation. In total four major problems are tackled using branch and bound. In each case one of the bounding schemes is applied, and extensive computational test results are presented. When appropriate, a comparison between previous and new algorithms shows the latter to be more efficient. The fourth and last part of the thesis gives a summary and suggests some promising lines of research that are worthy of investigation.