  After a brief outline of general aspects of conformal field theories in
coordinate space, in a first part we review the solution of the conformal
constraints of three- and four-point functions in momentum space in dimensions
$d\geq 2$, in the form of conformal Ward identities (CWI's). We center our
discussion on the analysis of correlators containing stress-energy tensors
$(T)$, conserved currents $(J)$, and scalar operators $(O)$. For scalar
four-point functions, we briefly discuss our method for determining the dual
conformal solutions of such equations, identified only by the CWI's, and
related to the conformal Yangian symmetry, introduced by us in previous work.
In correlation functions with $T$ tensors, evaluated around a flat spacetime,
the conformal anomaly is characterized by the (non-local) exchange of massless
poles in specific form factors, a signature that has been investigated both in
free field theory and non-perturbatively, by solving the conformal constraints.
We discuss the anomaly effective action, and illustrate the derivation of the
CWI's directly from its path integral definition and its Weyl symmetry, which
is alternative to the standard operatorial approach used in conformal field
theories in flat space. For two- and three-point functions, we elaborate on the
matching of these types of correlators to free-field theories. Perturbative
realizations of CFTs at one-loop provide the simplest expressions of the
general solutions identified by the CWI's, for generic operators $T$, $J$, and
scalars of specific scaling dimensions, by an appropriate choice of their field
content. In a technical appendix we offer details on the reconstruction of the
$TTO$ and $TTT$ correlators in the approach of Bzowski, McFadden and Skenderis,
and specifically on the secondary Ward identities of the method, in order to
establish a complete match with the perturbative description.
