This paper reformulates and extends the new, group theoretic, mobility criterion recently developed by the authors, (Rico, J. M., and Ravani, B., 2003, ASME J. Mech. Des., 125, pp. 70–80). In contrast to the Kutzbach-Grübler criterion, the new mobility criterion and the approach presented apply to a large class of single-loop overconstrained linkages. The criterion is reformulated, in terms of the well-known Jacobian matrices, for exceptional linkages and extended to linkages with partitioned mobility and trivial linkages. Several examples are included.