  The soft limits of scattering amplitudes have been extensively studied due to
their essential role in the computation of physical observables in collider
physics. The universal factorisation that occurs in these kinematic limits has
been shown to be related to conservation laws associated with asymptotic, or
large, gauge symmetries. This connection has led to a deeper understanding of
the symmetries of gauge and gravitational theories and to a reformulation of
scattering amplitudes in a basis of boost eigenstates which makes manifest the
two-dimensional global conformal symmetry of the celestial sphere. The recast,
or celestial, amplitudes possess many of the properties of conformal field
theory correlation functions which has suggested a path towards a holographic
description of asymptotically flat spacetimes. In this review we consider these
interconnected developments in our understanding of soft theorems, asymptotic
symmetries and conformal field theory with a focus on the structure and
symmetries of the celestial amplitudes and their holographic interpretation.
