  We investigate the phase structure of the SU(3) gauge theory with $N_f=8$ by
numerical simulations employing the massless Domain-Wall fermions.Our aim is to
study directly the massless quark region, since it is the most important region
to clarify the properties of conformal theories. When the number of flavor is
within the conformal window, it is claimed recently with Wilson quarks that
there is the conformal region at the small quark mass region in the parameter
space in addition to the confining phase and the deconfining phase. We study
the properties of the conformal region investing the spatial Polyakov loops and
the temporal meson propagators. Our data imply that there is the conformal
region, and a phase transition between the confining phase and the conformal
region takes place. These results are consistent with the claim that the
conformal window is between $7$ and $16$. Progress reports on other related
studies are also presented.
