The goal of this dissertation was to establish the relationship between a parameter descriptive of the trajectory of a smoothly curving crack, such as the curvature of the crack path, and the local stress state in the close vicinity of the crack tip. The behavior of fast -running cracks propagating along straight and smoothly curving paths in fracture specimens of various geometries was examined using dynamic photoelasticity and representations of the running crack stress field we redeveloped in terms of the coefficients of a set of infinite series, for both opening and shear mode loading conditions. Analysis of the isochromatic patterns, using local collocation methods based on this stress field representation, allowed the stress state in the neighborhood of the propagating crack-tip to be modelled with a high degree of accuracy and results were obtained for the variations with crack tip position of both the singular and leading non- singular stress field coefficients of interest. The results obtained for quasi-static and rapid crack propagation under opening mode conditions in a ring segment revealed the importance of retaining terms of order (at a minimum) r^1/2 even when only the singular term was to be determined accurately. Furthermore, it was found that the non-singular stress field coefficients varied similarly in both static and dynamic situations, with some variations in magnitude that could be attributed to crack speed. The results from the curved crack experiments also showed systematic variation of the non-singular terms, but more importantly, it was found that the instantaneous curvature of the crack path was related to the magnitude of the lowest order non-singular stress component (the coefficient of the r^1/2 term) associated with the local shear mode of deformation in the vicinity of the tip of the running crack. Furthermore, the results established that the only singularity associated with a crack propagating along a smoothly curving path in a brittle, isotropic material was that associated with the opening mode stress intensity factor, K1, and that the shear mode singularity, KII, was identically equal to zero.