Computing observables in conformal field theories (CFTs) in generic backgrounds and states represents an outstanding problem. In this thesis we develop a formalism to efficiently impose the kinematic constraints on the correlators of such theories based on the geometric construction of the ambient space by Fefferman and Graham. The latter is a Ricci-flat spacetime that can be thought of as a generalisation of the embedding space used for CFTs in vacuum and on conformally flat spaces. We test this formalism in the case of Euclidean thermal CFTs. We find perfect agreement with results from the thermal operator product expansion. We further produce novel holographic results for thermal scalar 2-point functions, which match the predictions of the ambient space formalism and provide new insight into both the analytic structure of these correlators and the role played by the double-twist spectrum. We then apply our formalism to CFTs on squashed spheres, generating new expressions for their scalar 2-point correlators. Finally, we establish connections of the ambient space with proposed approaches to flat holography and with the physics at spatial infinity in Beig-Schmidt gauge. By studying Einstein’s equations at spatial infinity we are able to prove the antipodal matching of the asymptotic BMS charges, a crucial assumption at the basis of a well-defined gravitational scattering problem in General Relativity and celestial holography.