One of the most difficult problems in designing composite structures is to ensure tolerance of severe damage. Current practice requires time-consuming and expensive testing to establish damage tolerance certification. Further, no predictive capability exists that allows assessment of the consequences for damage tolerance of changing the composite design, especially the orientation and thickness of plies in the lay-up, to optimize other performance characteristics. A new computational model of the evolution of severe damage is presented, that exploits some powerful new modeling concepts developed over the last 5 - 10 years. The model simulates delamination and splitting (shear) cracks, which grow in various orientations and change in shape with time; fiber rupture or microbuckling; global buckling of delaminated plies; and diffuse microcracking or shear damage within individual plies. A key feature of the model is the incorporation, in a finite element formulation, of elements that explicitly represent displacement discontinuities associated with cracks - so-called "cohesive elements." These elements relate the displacement discontinuity across a crack to the tractions that continue to act across the crack, which arise from polymer craze fibrils, bridging fibers, friction, and other possible origins. Properly chosen cohesive traction laws allow crack wake mechanisms to be represented with the correct physics. In typical severe damage problems, the crack process zones can be large and to ignore them would prohibit fidelity in modeling. Furthermore, the cohesive elements can be formulated in such a way that crack paths do not need to be prescribed in advance. The cohesive elements establish local failure conditions, which can be checked at any point in the structure at each computational increment - the crack paths evolve naturally. The formulation appears to be applicable to a wide variety of damage simulations. First results encourage the view that multiple damage mechanisms and thus the role of composite lay-up in damage tolerance can be predicted by simulations, reducing the reliance on expensive testing.