Analyses of fracture are discussed where the initial-boundary value problem formulation allows for the possibility of a complete loss of stress carrying capacity, with the associated creation of new free surface. No additional failure criterion is employed so that fracture arises as a natural outcome of the deformation process. Two types of analyses are reviewed. In one case, the material’s constitutive description incorporates a model of the failure mechanism; the nucleation, growth and coalescence of microvoids for ductile fracture in structural metals. In some analyses this is augmented with a simple characterization of failure by cleavage to analyze ductile-brittle transitions. The other class of problems involves specifying separation relations for one or more cohesive surfaces present in the continuum. The emphasis is on reviewing recent work on dynamic failure phenomena and the discussion centers around issues of length scales, size effects and the convergence of numerical solutions.