A variety of electronic phases in solid-state systems can be understood by abstracting away microscopic details and refocusing on how Fermi surface topology interacts with band structure to define available electron states 1 . In fact, topological concepts are broadly applicable to non-electronic materials and can be used to understand a variety of seemingly unrelated phenomena 2–6 . Here, we apply topological principles to origami-inspired mechanical metamaterials 7–12 , and demonstrate how to guide bulk kinematics by tailoring the crease configuration-space topology. Specifically, we show that by simply changing the crease angles, we modify the configuration-space topology, and drive origami structures to dramatically change their kinematics from being smoothly and continuously deformable to mechanically bistable and rigid. In addition, we examine how a topologically disjointed configuration space can be used to constrain the locally accessible deformations of a single folded sheet. While analyses of origami structures are typically dependent on the energetics of constitutive relations 11–14 , the topological abstractions introduced here are a separate and independent consideration that we use to analyse, understand and design these metamaterials. © 2018, The Author(s).