Abstract: Conformal geometry is strongly connected to Riemannian geometry in a very obvious way. A more subtle but very powerful link is provided by Fefferman-Graham's Poincare-Einstein manifold construction. Understanding the latter as a holonomy reduction of a Cartan geometry suggests new directions, natural generalisations of Riemannian geometry, and (for example) a nice variant of the projective geometry metricity question: Given a projective class $p$ of torsion free affine connections, is there a Levi-Civita connection in $p$? This picture draws on joint work with Andreas Cap, Matthias Hammerl, and Heather Macbeth.