  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 $SL(2,\mathbb C)$ Lorentz quantum numbers: namely their conformal
scaling dimension and spin $h\pm \bar h$ 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 $h=0$ or $\bar h=0$. For photons we explicitly construct
conformally soft $SL(2,\mathbb C)$ 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 $(2,0)$ $SL(2,\mathbb C)$ primary wavefunction.
Interestingly, BMS supertranslations are generated by a spin-one weight
$(\frac{3}{2},\frac{1}{2})$ operator, which nevertheless shares holomorphic
characteristics of a conformally soft operator. This is because the right hand
side of its OPE with a weight $(h,\bar h)$ operator ${\cal O}_{h,\bar h}$
involves the shifted operator ${\cal O}_{h+\frac{1}{2},\bar h+ \frac{1}{2}}$.
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.
