  We are taught that gauge transformations in classical and quantum mechanics
do not change the physics of the problem. Nevertheless here we discuss three
broad scenarios where under gauge transformations: (i) conservation laws are
not preserved in the usual manner; (ii) non-gauge-invariant quantities can be
associated with physical observables; and (iii) there are changes in the
physical boundary conditions of the wave function that render it
non-single-valued. We give worked examples that illustrate these points, in
contrast to general opinions from classic texts. We also give a historical
perspective on the development of Abelian gauge theory in relation to our
particular points. Our aim is to provide a discussion of these issues at the
graduate level.
