  We define and study asymptotic Killing and conformal Killing vectors in
$d$-dimensional Minkowski, (A)dS, $\mathbb{R}\times S^{d-1}$ and ${\rm
AdS}_2\times S^{d-2}$. We construct the associated quantum charges for an
arbitrary CFT and show they satisfy a closed algebra that includes the BMS as a
sub-algebra (i.e. supertranslations and superrotations) plus a novel
transformation we call `superdilations'. We study representations of this
algebra in the Hilbert space of the CFT, as well as the action of the finite
transformations obtained by exponentiating the charges. In the context of the
AdS/CFT correspondence, we propose a bulk holographic description in
semi-classical gravity that reproduces the results obtained from CFT
computations. We discuss the implications of our results regarding quantum
hairs of asymptotically flat (near-)extremal black holes.
