Origami has emerged as a potential tool for the design of mechanical metamaterials and metastructures whose novel properties originate from their crease patterns. Most of the attention in origami engineering has focused on the wellknown Miura-Ori, a folded tessellation that is flat-foldable for folded sheet and stacked blocks. This study advances the state of the art and expands the research field to investigate generic degree-4 vertex (4-vertex) origami, with a focus on facet-binding. In order to understand how facet-binding attributes to the mechanical properties of 4-vertex origami structures, geometries of the 4-vertex origami cells are analyzed and analytically expressed. Through repeating and stacking 4-vertex cells, origami sheets and stacked origami blocks can be constructed. Geometry analyses discover four mechanisms that will lead to the self-locking of 4-vertex origami cells, sheets, and stacked blocks: in-cell facet-binding, inlayer facet-binding, inter-layer facet binding, and in-layer and inter-layer facet-bindings. These mechanisms and the predicted self-locking phenomena are verified through 3D simulations and prototype experiments. Finally, this paper briefly introduces the unusual mechanical properties caused by the locking of 4-vertex origami structures. The research reported in this paper could foster a new breed of self-locking structures with various engineering applications.