<p>Solids subjected to repeated cycles of stress or deformation can fail after several cycles, a phenomenon termed fatigue failure. Although intensely investigated for a wide range of materials owing to its obvious practical importance, a microscopic understanding of the initiation of fatigue failure continues to be actively pursued, in particular for soft and amorphous materials. Employing computer simulations, here we show that upon approaching the so-called fatigue limit, the failure times of glasses subjected to cyclic shear deformation display a power-law divergence, which is at variance with commonly used functional forms, and exhibit a strong dependence on the degree of annealing of the glasses. Our simulations explore measures of damage based on a quantification of plastic rearrangements and on energy dissipated. The fraction of particles that undergo plastic rearrangements and the percolation transition they undergo are both predictive of failure. We also find a robust power law between the accumulated damage, which is quantified by the energy dissipated or the non-affine displacements, and the failure times, which permits prediction of failure times based on the behaviour in the initial cycles. These observations reveal salient new microscopic features of fatigue failure and indicate approaches for developing a full microscopic picture of fatigue failure in amorphous solids.</p>