Laminated composite structures have started to play a very significant role in today's aircraft industry. The application of composite materials has now gone beyond the borders of aircraft design and has entered into such fields as automotive, athletics and recreational equipment, etc. The light weight and high specific strength of composite material helps design vehicles with higher fuel efficiency and longevity. In order to understand the influence of design parameters related to the use of composite materials in these applications, a proper study of the laminated composite structures requires a complete failure analysis, which includes both initiation and propagation of damage. In this work a progressive failure methodology is developed and implemented in the commercial Finite Element software package, Abaqus. Out of the numerous failure criteria available in the literature to study damage initiation and propagation in unidirectional fiber reinforced composites, Puck and Schurmann's failure criteria have been chosen due to their ability to predict results close to those observed experimentally. Key features of the Puck and Schurmann's failure criteria for three-dimensional deformations of unidirectional fiber reinforced composites have been summarized. Failure modes in the matrix and the fiber are considered separately. The failure criteria are simplified for plane stress deformations. Whereas the failure plane can be analytically identified for plane stress deformations, a numerical search algorithm is needed for three-dimensional problems. Subsequent to the initiation of matrix failure, elastic moduli are degraded and values of these degradation parameters and fracture plane angles are found by using a Continuum Damage Mechanics (CDM) approach. It is found that the assumption that the material response remains transversely isotropic even after the matrix failure has initiated requires the degradation of the transverse Poisson's ratio. The Puck and Schurmann's failure criteria and the material degradation process have been implemented as a User Defined Field (USDFLD) subroutine in Abaqus. The implementation has been verified by analytically computing results for simple loadings and comparing them with predictions from using the USDFLD in Abaqus. Subsequently, both two- and three-dimensional problems of more realistic geometries and loadings have been analyzed and computed results compared with either experimental findings or results available in the literature. Major contributions of the work include identifying the degradation parameter for the transverse Poisson's ratio in terms of the matrix degradation parameter for the matrix failure in compression, development of the USDFLD based on Puck and Schurmann's failure criteria, implementing the USDFLD in the commercial finite element software, Abaqus, and verifying that results computing using the USDFLD for various laminates and loadings agree with those from either the analytical solution of the problem or those available in the literature. / Ph. D.