Single-loop N-bar linkages are closed kinematic chains that connect a number of rigid bars via revolute or prismatic joints in one loop. Over-constrained mechanisms are a special class of such linkages. In this class, the Grübler-Kutzbach-Chebyshev formula, computing the degrees-of-freedom (DoF) of the mechanism, fails and the causes of this failure are special geometric properties of the design parameters of the mechanism. Because of the low number of DoFs, single-loop linkages can be synthesized as simple single-purpose devices for specific tasks, such as pick-and-place operations. This can be advantageous compared to serial robots, because they can perform complicated motions following space-curves of higher degree with a low number of active joints.