  We present a gauge theory of the conformal group in four spacetime dimensions
with a non-vanishing torsion. In particular, we allow for a completely
antisymmetric torsion, equivalent by Hodge duality to an axial vector whose
presence does not spoil the conformal invariance of the theory, in contrast
with claims of antecedent literature. The requirement of conformal invariance
implies a differential condition on the aforementioned axial vector which leads
to a Maxwell-like equation in a four-dimensional curved background. We also
give some preliminary results in the context of $\mathcal{N}=1$
four-dimensional conformal supergravity in the geometric approach, showing that
if we only allow for the constraint of vanishing supertorsion all the other
constraints imposed in the spacetime approach are a consequence of the closure
of the Bianchi identities in superspace. This paves the way towards a future
complete investigation in the direction of a conformal theory of supergravity
with a non-vanishing (super)torsion.
