We present a strategy to design and construct self-assembling DNA nanostructures that define intricate curved surfaces in three-dimensional (3D) space using the DNA origami folding technique. Double-helical DNA is bent to follow the rounded contours of the target object, and potential strand crossovers are subsequently identified. Concentric rings of DNA are used to generate in-plane curvature, constrained to 2D by rationally designed geometries and crossover networks. Out-of-plane curvature is introduced by adjusting the particular position and pattern of crossovers between adjacent DNA double helices, whose conformation often deviates from the natural, B-form twist density. A series of DNA nanostructures with high curvature--such as 2D arrangements of concentric rings and 3D spherical shells, ellipsoidal shells, and a nanoflask--were assembled.