  We analyze the construction of conformal theories of gravity in the realm of
teleparallel theories. We first present a family of conformal theories which
are quadratic in the torsion tensor and are constructed out of the tetrad field
and of a scalar field. For a particular value of a coupling constant, and in
the gauge where the scalar field is restricted to assume a constant value, the
theory reduces to the teleparallel equivalent of general relativity, and the
tetrad field satisfies Einstein's equations. A second theory is formulated out
of the tetrad field only, and is not equivalent to the usual Weyl Lagrangian.
Therefore the latter is not the unique genuinely geometrical construction that
yields a conformally invariant action. The teleparallel framework allows more
possibilities for conformal theories of gravity.
