  Despite the fact that General Relativity (GR) has been very successful, many
alternative theories of gravity have attracted the attention of a significant
number of theoretical physicists. Among these theories, we have theories with
conformal symmetry. Here, the use of Weyl geometry to deal with conformal
teleparallel gravity is reviewed in great detail. As an application, a model
that can be set to be equivalent to the Teleparallel Equivalent of General
Relativity (TEGR) and is invariant under diffeomorphisms, local Lorentz
transformations (LLT) and Weyl transformations (WT) is created. Some $pp$-wave,
spherically symmetric and cosmological solutions are obtained. It turns out
that the class of possibles solutions is wider than that of TEGR. In addition,
the total and the gravitational energies of the universe are calculated and
analyzed.
