The initial value problem for a class of nonlinear hyperbolic systems is addressed in this paper, with initial data given in a Sobolev space of functions. These systems are not required to be strictly hyperbolic nor symmetrizable, and the admissible class includes systems with multiple characteristics. The fundamental requirement is a sign condition on the subprincipal operator of the linearized problem; this is parallel to a familiar condition for linear hyperbolic systems. Finally, the classical problem of water waves is shown to be related to this class of systems.