  We investigate the behavior of massless scalar, electromagnetic, and
linearized gravitational perturbations near null infinity in d \geq 4
dimensional Minkowski spacetime (of both even and odd dimension) under the
assumption that these fields admit a suitable expansion in 1/r. We also
investigate the behavior of asymptotically flat, nonlinear gravitational
perturbations near null infinity in all dimensions d\geq 4. We then consider
the memory effect in fully nonlinear general relativity. We show that in even
dimensions, the memory effect first arises at Coulombic order--i.e., order
1/r^{d-3}--and can naturally be decomposed into `null memory' and `ordinary
memory.' Null memory is associated with an energy flux to null infinity. We
show that ordinary memory is associated with the metric failing to be
stationary at one order faster fall-off than Coulombic in the past and/or
future. In odd dimensions, we show that the total memory effect at Coulombic
order and slower fall-off always vanishes. Null memory is always of `scalar
type', but the ordinary memory can be of any (i.e., scalar, vector, or tensor)
type. In 4-spacetime dimensions, we give an explicit example in linearized
gravity which gives rise to a nontrivial vector (i.e., magnetic parity)
ordinary memory effect at order 1/r. We show that scalar memory is described by
a diffeomorphism; vector and tensor memory cannot be. In d=4 dimensions, we
show that there is a close relationship between memory and the charge and flux
expressions associated with supertranslations. We analyze the behavior of
solutions that are stationary at Coulombic order and show how these suggest
`antipodal matching' between future and past null infinity, which gives rise to
conservation laws. The relationship between memory and infrared divergences of
the `out' state in quantum gravity is analyzed, and the nature of the `soft
theorems' is explained.
