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Novel insights into the asymptotic structure at null infinity suggest a holographic principle for spacetimes with (nearly) flat asymptotics: bulk scattering processes may be encoded in a lower-dimensional theory living on the celestial sphere at the null boundary of the spacetime. In this "celestial conformal field theory" the bulk S-matrix is recast as a conformal correlation function of Lorentz or boost eigentstates. Harnessing the power of CFT, I will explain how asymptotic symmetries at null infinity can be understood from the universal structure of soft theorems in quantum field theory together with conformal representation theory. Identifying the symmetries is a crucial first step in establishing the "celestial holography" program and I will discuss some of the recent developments in this direction.