  We introduce a new concept "conformal theories with an IR cutoff", after
pointing out that the following two categories in SU(3) gauge theories with
fundamental N_f fermions possess an IR fixed point: Large N_f QCD with N_f in
the conformal window (referred as Conformal QCD) and small N_f QCD with N_f out
of the conformal window at temperature T/Tc > 1 (referred as High Temperature
QCD).
  In the conformal theories with an IR cutoff there exists the "conformal
region'" together with the confining region and the deconfining region. We
verify numerically on a lattice of the size 16^3 x64 the existence of the
conformal region and the non-trivial Z(3) structure of the vacuum and the
Yukawa-type decay form of meson propagators in the conformal region.
  We stress that High Temperature QCD is intrinsically accompanied with an IR
cutoff. Therefore the understanding the vacuum structure and the property of
correlation functions is the key to resolve long standing issues in High
Temperature QCD.
  We further argue that there is a precise correspondence between Conformal QCD
and High Temperature QCD in the temporal propagators under the change of the
parameters N_f and T/T_c respectively: the one boundary is close to meson
states and the other is close to free quark states.
  In particular, we find the correspondence between Conformal QCD with N_f = 7
and High Temperature QCD with N_f=2 at T ~ 2 Tc being in close relation to a
meson unparticle model. From this we estimate the anomalous mass dimension
gamma* = 1.2 (1) for N_f=7.
