Four formulations of nonholonomic mechanical system dynamics, with both holonomic and differential constraints, are presented and shown to be well posed; i.e., solutions exist, are unique, and depend continuously on problem data. They are (1) the d'Alembert variational formulation, (2) a broadly applicable manifold theoretic extension of Maggi's equations that is a system of first-order ordinary differential equations (ODE), (3) Lagrange multiplier-based index 3 differential-algebraic equations (index 3 DAE), and (4) Lagrange multiplier-based index 0 differential-algebraic equations (index 0 DAE). The ODE formulation is shown to be well posed, as a direct consequence of the theory of ODE. The variational formulation is shown to be equivalent to the ODE formulation, hence also well posed. Finally, the index 3 DAE and index 0 DAE formulations are shown to be equivalent to the variational and ODE formulations, hence also well posed. These results fill a void in the literature and provide a theoretical foundation for nonholonomic mechanical system dynamics that is comparable to the theory of ODE.