Compliant mechanisms are flexible structures that utilize elastic deformation to achieve their desired motions. Using this unique mode of actuation, the compliant mechanisms have two distinct advantages over traditional rigid machines: (1) They can create highly repeatable motions that are critical for many high precision applications. (2) Their high degrees-of-freedom motions have the potential to achieve mechanical functionalities that are beyond traditional machines, making them especially appealing for miniature robots that are currently limited to only having simple rigid-body-motions and gripping functionalities. Unfortunately, despite the potential of compliant mechanisms, there are still several key challenges that restrict them from realizing their full potential. To facilitate this discussion, we first divide the compliant mechanisms into two categories: (1) the stiffer flexural mechanisms that are ideal for high precision applications, and (2) the more compliant miniature soft robots that can reshape their geometries to achieve highly complex mechanical functionalities. The key limitation for existing flexural mechanisms is that their stiffness and dynamic properties cannot be optimized when they have multi-degrees-of-freedom. This limitation has severely crippled the performance of flexural mechanisms because their stiffness and dynamic properties dictate their workspace, transient responses and capabilities to reject disturbances. On the other hand, miniature soft robots that have overall dimensions smaller than 1 cm, are unable to achieve their full potential because existing works do not have a systematic approach to determine the required design and control signals for the robots to generate their desired time-varying shapes.