In the present chapter, the motion equations of mechanical systems subject to either holonomic or nonholonomic constraints are derived not from variational principles, as is customary, but directly from the analysis of the restrictions imposed by the constraint equations on the acceleration of points in the system. We first consider in detail the constrained motion of one material point. Next, using the concept of a representative point, the above results are extended in a natural way to the problem of motion of a system of material points. They are further extended to mechanical systems consisting of material bodies. For this extension, we employ the concepts of a differentiable manifold and the tangent space to it.