  In the first part, we concentrate on CFTs in coordinate space. We lay the
foundations of Conformal Field Theory and we also demonstrate a method where by
using the embedding formalism we can derive up to n-point scalar conformal
correlators. We proceed with our analysis in momentum space and we illustrate
the theory of the conformal anomalies. We move on to analyse the
renormalization of the correlators through counterterms followed by a
discussion of the anomaly action. In the second part of the thesis. we derive
and analyze the conformal Ward identities (CWIs) of a tensorial 4-point
function of a generic CFT in momentum space. The correlator involves the
stress-energy tensor $T$ and three scalar operators $O$ ($TOOO$). We derive the
structure of the corresponding CWIs in two different sets of variables,
relevant for the analysis of the 1-to-3 (1 graviton $\to$ 3 scalars) and 2-to-2
(graviton + scalar $\to$ two scalars) scattering processes.
  Then, we move on to another conformal correlator, the one made of four
stress-energy tensors. We elaborate on the structure of the conformal anomaly
effective action up to 4-th order, in an expansion in the gravitational
fluctuations $(h)$ of the background metric, in the flat spacetime limit. We
discuss the renormalization of 4-point functions containing insertions of
stress-energy tensors (4T), in conformal field theories in four spacetime
dimensions.
  Finally, we include an analysis on the topological terms that are involved in
the renormalization and consequently corrections of gravitational theories.
