Origami has shown its potential in designing a three-dimensional folded structure from a flat sheet of material. In this paper, we present geometric design methods to construct cylindrical and axisymmetric origami structures that can fit between two given surfaces. Due to the symmetry of the structures, a strip of folds based on the generalized Miura-ori cells is first constructed and then replicated longitudinally/circumferentially to form the cylindrical/axisymmetric origami structures. In both designs, algorithms are presented to ensure that all vertexes are either on or strictly within the region between the target surfaces. The conditions of flat-foldability and developability are fulfilled at the inner vertexes and the designs are rigid-foldable with a single degree-of-freedom. The methods for cylindrical and axisymmetric designs are similar in implementation and of potential in designing origami structures for engineering purposes, such as foldcores, foldable shelters, and metamaterials.