Conforming materials to rigid substrates with Gaussian curvature-positive for spheres and negative for saddles-has proven a versatile tool to guide the self-assembly of defects such as scars, pleats, folds, blisters, and liquid crystal ripples. Here, we show how curvature can likewise be used to control material failure and guide the paths of cracks. In our experiments, and unlike in previous studies on cracked plates and shells, we constrained flat elastic sheets to adopt fixed curvature profiles. This constraint provides a geometric tool for controlling fracture behaviour: curvature can stimulate or suppress the growth of cracks and steer or arrest their propagation. A simple analytical model captures crack behaviour at the onset of propagation, while a two-dimensional phase-field model with an added curvature term successfully captures the crack's path. Because the curvature-induced stresses are independent of material parameters for isotropic, brittle media, our results apply across scales.