  The geometric description of gravitational memory for strong gravitational
waves is developed, with particular focus on shockwaves and their spinning
analogues, gyratons. Memory, which may be of position or velocity-encoded type,
characterises the residual separation of neighbouring `detector' geodesics
following the passage of a gravitational wave burst, and retains information on
the nature of the wave source. Here, it is shown how memory is encoded in the
Penrose limit of the original gravitational wave spacetime and a new `timelike
Penrose limit' is introduced to complement the original plane wave limit
appropriate to null congruences. A detailed analysis of memory is presented for
timelike and null geodesic congruences in impulsive and extended gravitational
shockwaves of Aichelburg-Sexl type, and for gyratons. Potential applications to
gravitational wave astronomy and to quantum gravity, especially infra-red
structure and ultra-high energy scattering, are briefly mentioned.
