• Weyl-covariant operators
• Weyl manifolds
• semiclassical gravity
• ambient metric
• trace anomaly
• Weyl symmetry

In the last decades, Conformal Field Theories (CFTs) have proven to be a very powerful framework in theoretical physics and Weyl symmetry constitutes the natural extension of conformal symmetry to curved spacetimes. In fact, in the context of renormalization, the fixed points of the RG flow often correspond to Weyl-invariant field theories, providing a structure of the space of theories. Moreover, the discovery of the Weyl or trace anomaly provided a series of questions and developements that find applications, as it is discussed throughout this thesis work, in possible extensions of the Standard Model of particles to curved spacetime. In this work, we discuss how we can exploit the trace anomaly to derive a semi-classical description of gravity in the presence of quantum matter fields. The trace anomaly can be expressed as the functional derivative of the effective action with respect to the metric: integrating this equation, we can infer the form of the effective action of the CFT in curved spacetime. Then, we introduce the Fefferman-Graham ambient metric formalism and its extension to Weyl manifolds to derive the expression of Weyl-covariant actions for fields of different spin. As a final application of our findings, we investigate how the addition of new Weyl-symmetric matter fields could result in the cancellation of the trace anomaly in the context of the Standard Model of particles coupled to a curved background.