The criterion for the plastic flow of crystalline solids is well established but a similar criterion for the fracture of brittle solids under triaxial stress conditions has not been proposed so far. According to the present theory, brittle fracture occurs as the result of the formation and subsequent propagation of microcracks. In this paper it is shown that the thermodynamic criterion for crack propagation is not a sufficient one and leads to unsatisfactory results in the general case. The necessary and sufficient criterion that there must be a local stress sufficient to rupture the atomic bonds at the edge of the crack does lead to satisfactory results. Griffith's crack is taken as a model and a calculation is carried out for the following boundary conditions: (1) at large distance from the crack there is an arbitrary plane stress or strain field; (2) at the crack boundary the crack surface is free from traction. This theory leads to a parabolic relationship τ 12 2 + 4K′τ22=4K′2 between the shear stress τ12 and the normal stress τ22 acting on the plane containing the propagating Griffith crack.