The variational methods of classical field theory may be applied to any theory with an action that is invariant under local gauge transformations. What is the significance of the resulting Noether current? This article examines such currents for both Abelian and non‐Abelian gauge theories and provides an explanation for their form and limited range of physical significance on a level accessible to those with a basic knowledge of classical field theory. Several of the more subtle aspects encountered in the application of the residual local gauge symmetry found by Becchi, Rouet, Stora, and Tyutin are also considered in detail in a self‐contained manner.