  Celestial scattering amplitudes for massless particles are Mellin transforms
of momentum-space scattering amplitudes with respect to the energies of the
external particles, and behave as conformal correlators on the celestial
sphere. However, there are few explicit cases of well-defined celestial
amplitudes, particularly for gravitational theories: the mixing between low-
and high-energy scales induced by the Mellin transform generically yields
divergent integrals. In this paper, we argue that the most natural object to
consider is the gravitational amplitude dressed by an oscillating phase arising
from semi-classical effects known as eikonal exponentiation. This leads to
gravitational celestial amplitudes which are analytic, apart from a set of
poles at integer negative conformal dimensions, whose degree and residues we
characterize. We also study the large conformal dimension limits, and provide
an asymptotic series representation for these celestial eikonal amplitudes. Our
investigation covers two different frameworks, related by eikonal
exponentiation: $2\to2$ scattering of scalars in flat spacetime and $1\to1$
scattering of a probe scalar particle in a curved, stationary spacetime. These
provide data which any putative celestial dual for Minkowski, shockwave or
black hole spacetimes must reproduce. We also derive dispersion and monodromy
relations for these celestial amplitudes and discuss Carrollian eikonal-probe
amplitudes in curved spacetimes.
