The four-dimensional S-matrix is reconsidered as a correlator on the celestial sphere at null infinity. Asymptotic particle states can be characterized by the point at which they enter or exit the celestial sphere as well as their SL(2, C) Lorentz quantum numbers: namely their conformal scaling dimension and spin h±h instead of the energy and momentum. This characterization precludes the notion of a soft particle whose energy is taken to zero. We propose it should be replaced by the notion of a conformally soft particle with h = 0 orh = 0. For photons we explicitly construct conformally soft SL(2, C) currents with dimensions (1, 0) and identify them with the generator of a U (1) Kac-Moody symmetry on the celestial sphere. For gravity the generator of celestial conformal symmetry is constructed from a (2, 0) SL(2, C) primary wavefunction. Interestingly, BMS supertranslations are generated by a spin-one weight ( 3 2 , 1 2 ) operator, which nevertheless shares holomorphic characteristics of a conformally soft operator. This is because the right hand side of its OPE with a weight (h,h) operator O h,h involves the shifted operator O h+ 1 2 ,h+ 1 2 . This OPE relation looks quite unusual from the celestial CFT 2 perspective but is equivalent to the leading soft graviton theorem and may usefully constrain celestial correlators in quantum gravity.