Direct methods are an alternative for power system transient stability analysis to avoid the enormous computational efforts of conventional time-domain method. The development of direct methods in last three decades makes it an effective potential approach to both on-line security assessment and off-line analysis tools. Among the direct methods, the Boundary of stability region based Controlling Unstable equilibrium point (BCU) method is the most successful, having a sound theoretical basis and practical application in power systems. It finds the controlling UEP of the original system via a reduced-state system. This thesis investigates the development of direct methods and the related theoretical foundation. Several widely used direct methods are presented and compared. The theoretical foundation and computational issues of BCU methods are discussed. Incorporation of more realistic power system models can be incorporated in BCU methods is introduced. Based on BCU method, some small system cases are tested for a given fault. / text