We study how geometry affects origami behavior and properties. Understanding origami from a structural standpoint can allow for conceptualizing and designing feasible applications across scales and disciplines of engineering. We review the basic mathematical rules of origami and use 3D-printed origami legos to illustrate those concepts. We then present an improved bar-and-hinge model to analyze the elastic stiffness, and estimate deformed shapes of origami – the model simulates three distinct behaviors: stretching and shearing of thin sheet panels; bending of flat panels; and bending along prescribed fold lines. We explore the stiffness of tubular origami and kirigami structures based on the Miura-ori folding pattern. A unique orientation for zipper coupling of rigidly foldable origami tubes substantially increases stiffness in higher order modes and permits only one flexible motion through which the structure can deploy. Deployment is permitted by localized bending along folds lines; however, other deformations are over-constrained (and engage the origami sheets in tension and compression). Furthermore, we couple compatible origami tubes into a variety of cellular assemblages including configurational metamaterials. We introduce origami tubes with polygonal cross-sections that can reconfigure into numerous geometries. The tubular structures satisfy the mathematical definitions for flat and rigid foldability, meaning that they can fully unfold from a flattened state with deformations occurring only at the fold lines. From a global viewpoint, the tubes do not need to be straight, and can be constructed to follow a non-linear curved line when deployed. From a local viewpoint, their cross-sections and kinematics can be reprogrammed by changing the direction of folding at some folds. The presentation concludes with a vision toward the field of origami engineering, including multifunctional origami, e.g. reconfigurable origami antennas.