  Fracture is a fundamental mechanism of materials failure. Propagating cracks
can exhibit a rich dynamical behavior controlled by a subtle interplay between
microscopic failure processes in the crack tip region and macroscopic
elasticity. We review recent approaches to understand crack dynamics using the
phase field method. This method, developed originally for phase
transformations, has the well-known advantage of avoiding explicit front
tracking by making material interfaces spatially diffuse. In a fracture
context, this method is able to capture both the short-scale physics of failure
and macroscopic linear elasticity within a self-consistent set of equations
that can be simulated on experimentally relevant length and time scales. We
discuss the relevance of different models, which stem from continuum field
descriptions of brittle materials and crystals, to address questions concerning
crack path selection and branching instabilities, as well as models that are
based on mesoscale concepts for crack tip scale selection. Open questions which
may be addressed using phase field models of fracture are summarized.
