A generalized theory of fatigue crack propagation is presented which is based on the linear accumulation of metallurgical damage in the plastic region. Fracture is considered to have occurred when the damage at the crack tip exceeds a critical value. An integral equation is derived which governs the growth of the fatigue crack. Successive analytic approximations to the solution of the integral equation are found and the corresponding errors estimated. The first approximation has a simple closed form and is shown to be accurate for a wide range of parameter values. A numerical solution of the integral equation confirms the accuracy of this approximation. The resulting crack growth law predicts a crack length at which catastrophic failure occurs.