In the paper, asymptotic methods, perturbation theory techniques, and their applications in nonlinear fracture mechanics are discussed. This review paper summarizes an overview of the asymptotic state of the art on the fracture behavior of nonlinear and damaged materials. The asymptotic stress, strain, and damage fields near the crack tip for power-law materials and the influence of the damage accumulation processes on the stress-strain state in the vicinity of the crack tip are analyzed. The paper gives a detailed review of the fundamental results obtained in nonlinear fracture mechanics by means of asymptotic methods and perturbation theory approaches. The main attention is paid to power-law materials and asymptotic stress and strain fields in the vicinity of the crack in both nondamaged materials and damaged materials under mixed-mode loadings. The paper analyzes the development of the asymptotic elastic-plastic crack-tip fields derived by Hutchinson, Rice, and Rosengren as a singular dominant term of the asymptotic expansion for the stress field in a power-law hardening material and shows the current state of the asymptotic methods and their applications in nonlinear fracture mechanics and continuum damage mechanics.