The possibility of an arbitrary conserved current, with any tensorial rank and having explicit dependence on space-time variables, being associated with an invariance property is investigated. It is proved, in the Lagrangian formalism of local quantum field theory, that conserved currents satisfying certain reasonable requirements must have the structure prescribed by Noether's theorem. It follows that the continuous group associated with the algebra of conserved current generators must be a symmetry group of the Lagrangian. The space-time dependence of the Bose part of the currents is restricted to be linear, and derivatives of Bose fields higher than the first are assumed to be absent. Making a distinction between invariance of the Lagrangian and that of the field equations, it is proved that an arbitrary conserved quantity has associated with it some invariance property of the field equations; no specific assumptions about the conserved currents or the field equations are required for this. All results are proved for interacting fields.