This dissertation presents new concepts and methodology in designing shape-morphing structures using bistable elements. Developed using the Pseudo-Rigid-Body Model (PRBM), linear bistable compliant mechanism elements produce predictable and controllable length changes. Step-by-step design procedures are developed to guide the design process of these bistable elements. Two different examples of Shape-Morphing Space Frames (SMSFs) were designed and prototyped utilizing the bistable linear elements in a single-layer grid, in addition to flexures and rigid links, to morph a cylindrical space frame into both a hyperbolic and a spherical space frame. Moreover, bistable unit-cell compliant-mechanism elements were also developed to morph a compact structure from a specific initial shape to a final specific shape. The detailed design of those unit cells were done using Computer-aided design (CAD) software following a novel design procedure to transform a one-degree-of-freedom mechanism into a structure with sufficient compliance within its links to toggle between two chosen stable positions. Two different design examples were investigated in this research and prototyped to demonstrate the ability to morph disks into a hemisphere or a sphere with the structure being stable in both states (disk and sphere).