We apply some recently developed control theoretic techniques to the analysis of a class of mechanical systems with constraints, Certain simple aspects of the theory of affine connections play an important part in our presentation, The necessary background is presented in order to illustrate how the methods may be applied. The bulk of this paper is devoted to a detailed analysis of some examples of nonholonomic mechanical control systems. We look at the Heisenberg system, the upright rolling disk, the roller racer, and the snakeboard.