  We study teleparallel gravitational theories with are invariant under the
conformal transformations. Wide family of the gravitational Lagrangians that
are invariant under conformal transformations have investigated. Cosmological
solutions inspired by the observational data for a flat Universe in the vacuum
has been found. To add matter fields to the cosmological models by preserving
the conformal symmetry there are significant limitations.
Friedmann-Robertson-Walker (FRW) equations based on the effective terms of
energy density and pressure have been rewriting and continuity equation for the
effective quantities are derived. In the vacuum and in the absence of any
matter field,the FRW equation has an exact solution for Hubble parameter which
is consistent with the cosmological data,specially given analytical solution is
in good agreement with $\Lambda$CDM model in the present time. Then the scalar
field-Torsion models in the cosmological FRW background investigated. Massless
scalar field equations are very complex with an exact analytical solution in
special limits. We have shown that the Lagrangian scalar field with self
interaction $V(\phi)=\frac{1}{4!}\mu\phi^4$ can be written that the conformal
symmetry is preserved.
