Impacts into solids are invariably accompanied by the abundant formation of cracks and fragments created by crack linkage. In contrast to the dominantly compressive nature of impact-generated stress waves, these cracks form in a tensile stress regime created either by reflection of the compressive waves from free surfaces or by displacement of material outward from the impact site. The formation of cracks is thus a complex process, which is also highly nonlinear because crack formation strongly affects the propagation of subsequent stress waves. In this paper we generalize a continuum damage model of dynamic fragmentation originally proposed by Grady and Kipp to two and higher dimensions. We develop algorithms that permit efficient computer implementation of this model in the context of a Lagrangian hydrocode and compare the code predictions to an extensive suite of laboratory impact fragmentation experiments. We find that both the largest fragment size and many details of the fragment size-number distribution are faithfully reproduced by the code, including the previously enigmatic segmentation of the cumulative size-number distributions. We also argue that the mode of failure may be different for laboratory size scales and geologically interesting problems such as multikilometer-scale impact cratering or asteroid fragmentation, making it imperative to use physical modeling rather than empirical scaling laws to address fragmentation at large size scales.