We study the geometric mechanics of origami assemblages and investigate 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.