The concept of polyhedra with articulated faces is investigated in this paper. Polyhedra with articulated faces can be described as polyhedral frameworks, whose faces are constrained to remain planar. A mechanical arrangement based on a single type of component is proposed for the construction of the polyhedra. Then, the determination of their infinitesimal and full-cycle mobility is addressed. In some cases, they are rigid structures while in others they are articulated mechanisms. Finally, examples are given, using simulation and physical models, and several new families of articulated polyhedra are synthesized.