Fracture is one of the most prominent concerns for large scale applications of graphene. In this paper, we review some of the recent progresses in experimental and theoretical studies on the fracture behaviors of graphene, with discussions touching theoretical strength, mode I fracture toughness, mixed mode fracture, chemical fracture, irradiation fracture, dynamic fracture, impact fracture, and sonication fracture. In spite of rapid developments in experiments and simulations, there are still significant yet unresolved issues related to the fracture of graphene, examples including: (1) Can one enhance the toughness of graphene with designed topological defects? (2) How does grain size affect the strength of polycrystalline graphene? (3) How do the out-of-plane effects (e.g., wrinkle caused by external loading or curvature induced by topological defects) influence the fracture of graphene? (4) Can one develop a continuum model with the ability to capture graphene fracture with complicated modes, such as shear fracture coupled with wrinkling deformation and tear fracture? (5) How does fracture occur when tearing a polycrystalline graphene sheet? (6) Can one control the fracture behavior of graphene by combing the chemical, irradiation and stress effect? (7) How fast can cracks propagate in graphene? (8) What is the behavior of interfacial cracks in graphene, i.e., cracks along the grain boundaries or interfaces of heterogeneous structures? (9) How does a multilayer graphene membrane break under high speed impact and why such structures can absorb a large amount of kinetic energy? (10) Can one tailor/design the graphene structures with controlled fracture? The intention here is not to provide complete answers to such questions, but to draw attention from the mechanics community to them as potential research topics.