In this contribution we have considered how the mathematical description of cracks was changing. A sketch of the works done in the framework of the theory elasticity and fracture mechanics that have relation to stress concentration around cracks is given. Considerable attention was focused on the physical ideas introduced into the theory of elasticity and fracture mechanics by Griffith and Mott. A sketch of the new approach, molecular dynamics simulations, is also given. It is shown that this approach revealed some features of crack behavior in crystalline and amorphous materials, polymers, and carbon nanotubes that cannot be imagined on the basis of speculative knowledge. At last,we have tried to attract attention to using the theory of solitons for solving nonlinear problems of the theory of elasticity and fracture mechanics. The solitonic theory of a moving crack is thoroughly discussed.