The paper presents a three-dimensional finite-deformation cohesive element and a class of irreversible cohesive laws designed to accurately and efficiently track dynamically growing cracks. The cohesive element models the separation of crack flanks according to an irreversible cohesive law, leading to the formation of free surfaces, and is compatible with conventional finite element discretization of the bulk material. The authors extend the cohesive model of Camacho and Ortiz to three dimensions, assuming that cohesive surfaces unload to the origin and mode coupling is accounted for by an effective scalar opening displacement. The cohesive elements are endowed with full finite kinematics and are compatible with tetrahedral elements. The versatility and predictive ability of the method are demonstrated through a simulation of a drop-weight dynamic fracture test, where the method accurately captures the crack-tip trajectory and other salient features of the experimental record.The paper presents a three-dimensional finite-deformation cohesive element and a class of irreversible cohesive laws designed to accurately and efficiently track dynamically growing cracks. The cohesive element models the separation of crack flanks according to an irreversible cohesive law, leading to the formation of free surfaces, and is compatible with conventional finite element discretization of the bulk material. The authors extend the cohesive model of Camacho and Ortiz to three dimensions, assuming that cohesive surfaces unload to the origin and mode coupling is accounted for by an effective scalar opening displacement. The cohesive elements are endowed with full finite kinematics and are compatible with tetrahedral elements. The versatility and predictive ability of the method are demonstrated through a simulation of a drop-weight dynamic fracture test, where the method accurately captures the crack-tip trajectory and other salient features of the experimental record.