  The symmetries of the gravitational scattering are intimately tied to the
symmetries which preserve asymptotic flatness at null infinity. In Penrose's
definition of asymptotic flatness, a central role is played by the notion of
asymptotic simplicity and the ensuing peeling behavior which dictates the decay
rate of the Weyl tensor. However, there is now accumulating evidence that in a
generic gravitational scattering the peeling property is broken, so that the
spacetime is not asymptotically-flat in the usual sense. These obstructions to
peeling can be traced back to the existence of universal radiative low
frequency observables called ''tails to the displacement memory''. The
universality of these tail modes is the statement of the classical logarithmic
soft graviton theorem of Sahoo, Saha and Sen. Four-dimensional gravitation
scattering therefore exhibits a rich infrared interplay between tail to the
memory, loss of peeling, and universal logarithmic soft theorems. In this paper
we study the solution space and the asymptotic symmetries for
logarithmically-asymptotically-flat spacetimes. These are defined by a
polyhomogeneous expansion of the Bondi metric which gives rise to a loss of
peeling, and represent the classical arena which can accommodate a generic
gravitational scattering containing tails to the memory. We show that while the
codimension-two generalized BMS charges are sensitive to the loss of peeling at
$\mathcal{I}^+$, the flux is insensitive to the fate of peeling. Due to the
tail to the memory, the soft superrotation flux contains a logarithmic
divergence whose coefficient is the quantity which is conserved in the
scattering by virtue of the logarithmic soft theorem. In our analysis we also
exhibit new logarithmic evolution equations and flux-balance laws, whose
presence suggests the existence of an infinite tower of subleading logarithmic
soft graviton theorems.
