  Conformal symmetry always played an important role in field theory (both
quantum and classical) and in gravity. We present construction of quantum
conformal gravity and discuss its features regarding scattering amplitudes and
quantum effective action. First, the long and complicated story of
UV-divergences is recalled. With the development of UV-finite higher derivative
(or non-local) gravitational theory, all problems with infinities and spacetime
singularities are solved. Moreover, the non-local quantum conformal theory
reveals itself to be ghost-free, so the unitarity of the theory is safe. After
the construction of UV-finite theory, we focused on making it manifestly
conformally invariant using the dilaton trick. We also argue that in this class
of theories conformal anomaly vanishes by fine-tuning the couplings. As
applications of this theory, the constraints of the conformal symmetry on the
form of the effective action and on the scattering amplitudes are shown. We
also remark about the preservation of the unitarity bound for scattering.
Finally, the old model of conformal supergravity by Fradkin and Tseytlin is
briefly presented.
