Abstract: In this paper, the K criterion for linear fracture mechanics, the G criterion for interfacial fracture mechanics, and the J criterion for elastoplastic fracture mechanics has been briefly reviewed, from the preliminary works of Inglis and Griffith to the foundation contributions of Irwin and Rice. Recently, on the basis of the G criterion for interfacial fracture mechanics, the K criterion for interfacial fracture mechanics was derived. This shows that the criteria of fracture mechanics need to be further improved. Thereupon, some debatable issues in the criteria of fracture mechanics are raised. Then, the strain discontinuity at the crack tip is assumed as the source of the stress singularity at a crack tip, and the physical meaning of the stress intensity factor is discussed. At the end, it is concluded that an improved and reliable analysis for the elastoplastic stress field near a crack tip is the key for the establishment of a criterion of elastoplastic fracture mechanics, and an improved and reliable analysis for the elastoplastic stress field near a crack tip depends on the determination of the magnitude of the stress at the crack tip, which caused the strain discontinuity at the crack tip.