In order to arrive at safe and reliable design of composite structures, understanding of the mechanisms and mechanics of damage growth in these materials is of paramount significance. Numerical models, if designed, implemented and used carefully, can be helpful not only to understand the mechanisms and mechanics of damage growth but also to predict the susceptibility of a structure to failure. In this thesis, advanced finite elements and numerical methods are explored to develop an integrated, computationally efficient and reliable numerical framework for the analysis of interacting damage mechanisms in laminated composite plates/shells subjected to transverse quasi-static and dynamic loading. A solid-like shell element is used to obtain a three-dimensional stress state in fiber-reinforced laminated composites. The element is further extended to model mesh-independent matrix cracking by incorporating a discontinuity in the shell mid-surface, shell director and thickness stretching field using the phantom node method. A progressive failure model is developed which is able to simulate impact induced damage in laminated composites. Care is taken to accurately describe the interaction between matrix cracks and delamination damage which is crucial for accurate predictions of fracture phenomena and laminate strength. Furthermore, a time-dependent progressive failure model is developed to simulate crack growth in laminated composites under dynamic loading conditions. The proposed mass discretization schemes for the solid-like shell element ensure efficient performance of the element in implicit as well as explicit elasto-fracture analysis of composite laminates. The presented numerical framework also discusses computational modeling of coupled thermo-mechanics of laminated composites in the presence of cracks. The unified computational model is able to simulate coupled adiabatic-isothermal cracks propagating arbitrarily through the finite element mesh.