Bistable and multistable structures have attracted significant attention in recent years due to their ability to maintain multiple stable equilibrium states without continuous external energy. These characteristics make them highly suitable for designing adaptive, programmable, and intelligent structures. The advent of additive manufacturing technologies, particularly (FFF), has created new opportunities for realizing such structures by enabling precise control over material anisotropy, thermal residual stresses, pre-strain, and structural geometry. This review paper presents a systematic overview of the fundamental principles and methods for achieving bistability and multistability through 3D printing. The reviewed approaches include the intentional use of thermal residual stresses, pre-strained layers, geometry-driven bistable mechanisms, bio-inspired designs, origami and kirigami concepts, as well as reinforced and composite filaments. Recent studies demonstrate that by programming extrusion paths, layer orientation, thermal history, and material distribution, the potential energy landscape of printed structures can be effectively engineered, allowing accurate control over the number and stability of equilibrium states. These capabilities have enabled applications in soft robotics, deployable aerospace structures, biomedical devices, energy harvesting systems, and adaptive metamaterials. Finally, key research challenges are discussed, including the prediction of stable states, dependence of mechanical behavior on printing parameters, time-dependent material effects such as creep and fatigue, and the lack of comprehensive analytical design models. Future research directions toward developing next-generation programmable multistable structures enabled by additive manufacturing are also outlined.