A consistent relativistic formalism is developed for electron-nucleus scattering. It is based on renormalizable relativistic quantum field theories of mesons and baryons (quantum hadrodynamics). The relativistic Hartree approximation describes the target using four-component Dirac wave functions. An effective, local electromagnetic current operator is constructed that reproduces electron scattering from a single nucleon. This operator is explicitly conserved in the relativistic Hartree approximation and, together with the Hatree wave functions, provides a consistent covariant formalism for electron-nucleus scattering. Applications of the formalism to elastic charge and magnetic scattering are discussed.