Bistable mechanical systems having two local minima of potential energy can rest in either of the two stable equilibrium states in the absence of external loadings. A snap-through action may occur under suitable stimuli and/or loading, during which such systems exhibit distinct properties from linear structures. Such kinds of structures have been widely exploited for designing advanced functional systems for a variety of applications. Here, the advances of bistable structures are summarized for novel advanced functional systems, including actuators, energy harvesters, microelectromechanical systems (MEMS), robotics, energy absorbers, and programmable devices as well as metamaterials. The controllable snap-through motions are highlighted in the nonlinear structures of bistability/multistability. Finally, the major principles, structures, pros and cons, and the future research directions along with its challenges are discussed.