Conformal symmetry has surfaced in many different areas of gravitational physics, notably in the AdS/CFT correspondence, Horava gravity, Shape Dynamics and the initial value problem in GR, and may very well be our most important clue to a UV-completion of gravity. On the other hand, the early universe offers observational evidence for conformal symmetry in the form of a near-scale invariant primordial power spectrum, as seen in the cosmic microwave background and large-scale structure of the universe. In particular, the trading of symmetries between spacetime refoliation invariance for spatial conformal invariance, as in Shape Dynamics, may be manifested in the scale-invariant dynamics of the early universe. This might also lead to a violation of Lorentz symmetry in the gravitational sector, as in Einstein-Aether or Cuscuton gravity. This conference brings researchers in these areas together to foster new interdisciplinary discourse and collaboration in this fascinating interplay between conformal symmetry, cosmology and gravitational physics.