  Carrollian holography aims to express gravity in four-dimensional
asymptotically flat spacetime in terms of a dual three-dimensional Carrollian
CFT living at null infinity. Carrollian amplitudes are massless scattering
amplitudes written in terms of asymptotic or null data at $\mathscr I$. These
position space amplitudes at $\mathscr I$ are to be re-interpreted as
correlation functions in the putative dual Carrollian CFT. We derive basic
results concerning tree-level Carrollian amplitudes yielding dynamical
constraints on the holographic dual. We obtain surprisingly compact expressions
for $n$-point MHV gluon and graviton amplitudes in position space at $\mathscr
I$. We discuss the UV/IR behaviours of Carrollian amplitudes and investigate
their collinear limit, which allows us to define a notion of Carrollian OPE. By
smearing the OPE along the generators of null infinity, we obtain the action of
the celestial symmetries - namely, the $S$ algebra for Yang-Mills theory and
$Lw_{1+\infty}$ for gravity - on the Carrollian operators. As a consistency
check, we systematically relate our results with celestial amplitudes using the
link between the two approaches. Finally, we initiate a direct connection
between twistor space and Carrollian amplitudes.
