In this paper the authors treat elasticity as a field theory derived from a Lagrangian. The basic fields are maps from spacetime to a three-dimensional “material manifold” \(B\). Objects defined on \(B\) are physically interpreted as properties of the material prior to the action of deformations or other fields. The field equations are cast into a first-order symmetric hyperbolic system. As a consequence they obtain local-in-time existence and uniqueness theorems under various circumstances.