  We consider the construction of gauge theories of gravity that are invariant
under local conformal transformations. We first clarify the geometric nature of
global conformal transformations, in both their infinitesimal and finite forms,
and the consequences of global conformal invariance for field theories, before
reconsidering existing approaches for gauging the conformal group, namely
auxiliary conformal gauge theory and biconformal gauge theory, neither of which
is generally accepted as a complete solution. We then demonstrate that,
provided any matter fields belong to an irreducible representation of the
Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be
considered as an alternative method for gauging the conformal group, since eWGT
is invariant under the full set of local conformal transformations, including
inversions, as well as possessing conservation laws that provide a natural
local generalisation of those satisfied by field theories with global conformal
invariance, and also having an `ungauged' limit that corresponds to global
conformal transformations. By contrast, although standard Weyl gauge theory
also enjoys the first of these properties, it does not share the other two, and
so cannot be considered a valid gauge theory of the conformal group.
