A methodology for designing cylindrical origami with different patterns and different cross-sectional shapes is presented. Planar and cylindrical origami with curved-crease patterns are designed and proved to be foldable and developable regardless of the type of crease line. Specifically, cylindrical origami structures with arc-Miura pattern and circular cross section, arc pattern and circular cross section, arc-Miura pattern and triangular cross section, arc pattern and triangular cross section are designed. The relationships between the two-dimensional (2D) crease lines in a sheet and three-dimensional (3D) spatial origami structures are derived. Fabrications of cylindrical origami by a folding process are performed to validate the design theory for 2D crease lines. This work paves the way for applications for cylindrical origami in engineering, such as energy absorption.