Novel insights into quantum gravity in asymptotically flat spacetimes evolving around soft theorems in scattering amplitudes, memory effects and asymptotic symmetries hint at an underlying holographic structure of Minkowski spacetime: information about 4D quantum gravity might be encoded in a 2D CFT on the celestial sphere at the conformal boundary of Minkowski spacetime. I will discuss recent progress on this attempted formulation of a flat space holography focusing on the 4D S-matrix which takes the form of a 2D correlator on the celestial sphere in a conformal basis. I will discuss how celestial conformal symmetry is generated by "conformally soft" gravitons and how insertions of the BMS supertranslation current in a correlator gives rise to the celestial analogue of Weinberg's soft graviton theorem.