This paper for the first time presents an overconstrained spatial eight-bar linkage and its application to the synthesis of a group of Fulleroid-like deployable Platonic mechanisms. Structure of the proposed eight-bar linkage is introduced, and constrain and mobility of the linkage are revealed based on screw theory. Then by integrating the proposed eight-bar linkage into Platonic polyhedron bases, synthesis of a group of Fulleroid-like deployable Platonic mechanism is carried out and illustrated by the synthesis and construction of a Fulleroid-like deployable tetrahedral mechanism. Further, mobility of the Fulleroid-like deployable Platonic mechanisms is formulated via constraint matrices by following Kirchhoff's circulation law for mechanical networks, and kinematics of the mechanisms is presented with numerical simulations illustrating the intrinsic kinematic properties of the group of Fulleroid-like deployable Platonic mechanisms. In addition, a prototype of the Fulleroid-like deployable spherical-shape hexahedral mechanism is fabricated and tested verifying mobility and kinematic characteristics of the proposed deployable polyhedral mechanisms. Application of the proposed deployable Platonic mechanisms is demonstrated in the development of a transformable quadrotor. This paper hence presents a novel overconstrained spatial eight-bar linkage and a new geometrically intuitive method for synthesizing Fulleroid-like regular deployable polyhedral mechanisms that have great potential applications in deployable, recongurable, and multi-functional robots.