We recast 4D scattering amplitudes and their soft limits as correlators of a 2D CFT on the celestial sphere. Our construction relies on a foliation of 4D flat space into a family of 3D hyperbolic geometries to which the AdS3/CFT2 dictionary is directly applicable. By reformulating 4D scattering amplitudes as 3D Witten diagrams dual to 2D correlators, we show how the Ward identities of the 2D CFT are equivalent to the 4D soft theorems. Moreover, we demonstrate how the infinite-dimensional Kac-Moody and Virasoro algebras of the 2D CFT are manifested as the asymptotic symmetries of 4D flat space. Finally, we discuss the interpretation of 4D electromagnetic and gravitational memory effects as a certain version of the 3D Aharonov-Bohm effect.