We show that the simplest building blocks of origami-based materials-rigid, degree-four vertices-are generically multistable. The existence of two distinct branches of folding motion emerging from the flat state suggests at least bistability, but we show how nonlinearities in the folding motions allow generic vertex geometries to have as many as five stable states. In special geometries with collinear folds and symmetry, more branches emerge leading to as many as six stable states. Tuning the fold energy parameters, we show how monostability is also possible. Finally, we show how to program the stability features of a single vertex into a periodic fold tessellation. The resulting metasheets provide a previously unanticipated functionality-tunable and switchable shape and size via multistability. [14,15]. The building blocks for these materials are typically quasi-1D rods or springs, but recently, origami-inspired metamaterials made from folding planar structures have gained interest. This represents an important departure for a variety of reasons. First, the deformations of folding-based materials can be highly nonlinear owing to the complex constraint space imposed by the fold network. Second, their energetic landscapes do not arise from central-force linear springs but instead through torsional spring interactions [16][17][18][19].Most recent attention has been focused on the Miura-Ori, a fold tessellation well known for its negative Poisson's ratio. Silverberg et al. recently used Miura-Ori to create a metamaterial with tunable stiffness by introducing a reversible "pop-through" defect [18]. This local defect, permitted via plate bending, is one of a few specific examples of bistability in folding planar structures-others include the symmetric water bomb vertex [20] and the hypar [21]. Such multistability is a desirable property for the design of metamaterials as it allows reprogrammable reconfiguration of shape and bulk properties.Here, we reveal how folding planar structures offer a platform for globally multistable metamaterials-structures capable of multiple stable shapes and sizes. Our building block is the degree-four vertex, i.e., four rigid plates connected by four folds (or hinges) that meet at a point. This is the simplest building block for origami metamaterials because it is a one-degree-of-freedom mechanism (lower n-vertices are rigid [22]). We show that the interesting physical properties arise from complexity in the physical configuration space, to which we now turn our attention.Generic configuration space.-We first consider generic four-vertices, i.e., those without collinear folds, symmetry, or flat foldability [23]. We specify the flat-state geometry by the set of sector angles fα i g, where each α i < π and