The familiar classical dynamical theory treats only a restricted class of dynamical systems. For a Lagrangian depending on n generalized coordinates, the n generalized velocities, and (perhaps) the time the usual theory considers only the normal case in which the Hessian matrix of the Lagrangian with respect to the generalized velocities has rank n. But the rank of this Hessian matrix can be less than n in a given dynamical system. Dynamical systems with this property are called degenerate in the present work. Some important physical systems do in fact belong to this degenerate class: for instance, Einstein's theory of the gravitational field, the theory of the electromagnetic field in terms of the electromagnetic potentials, and the theory of the neutral vector meson field. This circumstance makes it necessary to have a general theory of degenerate dynamical systems (and of the associated degenerate variational problems).