A general formulation by dual boundary integral equations and a computational solution algorithm for the general mixed-mode crack in a linearly elastic, isotropic medium is presented. Traction boundary integral equations are collocated at the points on one crack surface while displacement boundary integral equations are collocated at the opposite points on the other surface to ensure a unique solution. The hypersingular and strongly singular integrals in the traction and displacement boundary integral equations are regularized before the numerical implementation. The singular integration elements on both crack surfaces are replaced by smooth curved auxiliary surfaces to avoid the direct integration over the singular elements. Usage of these detoured auxiliary contours is justified by certain identities of the fundamental solutions. Convergence tests for integration order and subdivision of the auxiliary surface elements were performed. To demonstrate the accuracy and efficiency of the present technique, the deformation and the stress intensity factors for two- and three-dimensional embedded and edge crack problems are given.