Single-pass cutting of glass substrates with a curved edge is demonstrated using ultrafast curving Bessel and Airy beams, and the advantages of each beam are considered. To achieve successful cutting using the curving Bessel beam, crack directionality is controlled to be perpendicular to the cut by modifying the beam’s spectrum, and a flat-top intensity profile is used to increase consistency. When cutting using Airy beams, a modified beam called an oblong Airy beam is used, which has an oblong focal-spot shape, flat-top intensity profile, and increased contrast between the focal spot and sidelobes as compared to a Gaussian Airy beam. In both cases, the beams’ phase profiles are chosen to maximize edge curvature for a specific sample thickness. Cutting of 500 μm thick glasses in a single laser pass with a high (>10 μm) pitch between laser shots is demonstrated.