Nonholonomic mechanical systems are revisited. This review article focuses on Lagrangian formulations leading to a system of governing equations free of constraint forces. While eliminating the constraint forces, the number of scalar Lagrange equations is reduced to a number of independent equations lower than the original system with constraint forces. In the process of constraint-force elimination and dimension-reduction, a matrix that appears to play a relevant role in the formulation of the mathematical models of mechanical systems arises naturally. We call this matrix here the holonomy matrix. It is shown that necessary and sufficient conditions for the integrability of the constraints in Pfaffian form are readily derived using the holonomy matrix. In the same vein, a class of nonholonomic systems is identified, of current engineering relevance, that is termed quasiholonomic. Examples are included to illustrate these concepts. This review article contains 40 references.