  The free Maxwell theory in D<>4 dimensions provides a physical example of a
unitary, scale invariant theory which is NOT conformally invariant. The easiest
way to see this is that the field strength operator F_mn is neither a primary
nor a descendant. We show how conformal multiplets can be completed, and
conformality restored, by adding new local operators to the theory. In D>=5,
this can only be done by sacrificing unitarity of the extended Hilbert space.
We analyze the full symmetry structure of the extended theory, which turns out
to be related to the OSp(D,2|2) superalgebra.
