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After a brief outline of general aspects of conformal field theories incoordinate space, in a first part we review the solution of the conformalconstraints 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 ourdiscussion on the analysis of correlators containing stress-energy tensors$(T)$, conserved currents $(J)$, and scalar operators $(O)$. For scalarfour-point functions, we briefly discuss our method for determining the dualconformal solutions of such equations, identified only by the CWI's, andrelated 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 masslesspoles in specific form factors, a signature that has been investigated both infree field theory and non-perturbatively, by solving the conformal constraints.We discuss the anomaly effective action, and illustrate the derivation of theCWI's directly from its path integral definition and its Weyl symmetry, whichis alternative to the standard operatorial approach used in conformal fieldtheories in flat space. For two- and three-point functions, we elaborate on thematching of these types of correlators to free-field theories. Perturbativerealizations of CFTs at one-loop provide the simplest expressions of thegeneral solutions identified by the CWI's, for generic operators $T$, $J$, andscalars of specific scaling dimensions, by an appropriate choice of their fieldcontent. 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 toestablish a complete match with the perturbative description.

Conformal field theory in momentum space and anomaly actions in gravity:The analysis of three- and four-point functions