Origami and paperfolding techniques may inspire the design of structures that have the ability to be folded and unfolded: their geometry can be changed from an extended, servicing state to a compact one, and back-forth. In traditional Origami, folds are introduced in a sheet of paper (a developable surface) for transforming its shape, with artistic or decorative intent; in recent times the ideas behind origami techniques were transferred in various design disciplines to build developable foldable / unfoldable structures, mostly in aerospace industry. The geometrical arrangement of folds allows a folding mechanism of great efficiency and is often derived from the buckling patterns of simple geometries, like a plane or a cylinder (e.g. Miura-Ori and Yoshimura folding pattern). Here we interest ourselves to the conception of foldable / unfoldable structures for civil engineering and architecture. In those disciplines, the need for folding efficiency comes along with the need for structural efficiency (stiffness); for this purpose we will explore nondevelopable foldable / unfoldable structures: those structures exhibit potential stiffness because, when unfolded, they cannot be flattened to a plane (non-developability). In this paper we propose a classification for foldable / unfoldable surfaces that comprehend non fully developable (and also non fully foldable) surfaces and a method for the description of folding motion. Then we propose innovative geometrical configurations for those structures by generalizing the Miura-Ori folding pattern to non-developable surfaces that, once unfolded, exhibit curvature.