  Scattering amplitudes in $d+2$ dimensions can be expressed in terms of a
conformal basis, for which the S-matrix behaves as a CFT correlation function
on the celestial $d$-sphere. We explain how compact expressions for the full
tree-level S-matrix of gauge theory, gravity and other QFTs extend to this
conformal basis, and are easily derived from ambitwistor strings. Using these
formulae and their worldsheet origins, we prove various tree-level 'conformal
soft theorems' in gauge theory and gravity in any dimension; these arise from
limits where the scaling dimension of an external state in the scattering
process takes special values. These conformally soft limits are obscure from
standard methods, but they are easily derived with ambitwistor strings.
Additionally, we make an identification between the residues of conformally
soft vertex operator insertions in ambitwistor strings and charges generating
asymptotic symmetries.
