  We review Rydberg aggregates, assemblies of a few Rydberg atoms exhibiting
energy transport through collective eigenstates, considering isolated atoms or
assemblies embedded within clouds of cold ground-state atoms. We classify
Rydberg aggregates, and provide an overview of their possible applications as
quantum simulators for phenomena from chemical or biological physics. Our main
focus is on flexible Rydberg aggregates, in which atomic motion is an essential
feature. In these, simultaneous control over Rydberg-Rydberg interactions,
external trapping and electronic energies, allows Born-Oppenheimer surfaces for
the motion of the entire aggregate to be taylored as desired. This is
illustrated with theory proposals towards the demonstration of joint motion and
excitation transport, conical intersections and non-adiabatic effects.
  Additional flexibility for quantum simulations is enabled by the use of
dressed dipole-dipole interactions or the embedding of the aggregate in a cold
gas or BEC environment. Finally we provide some guidance regarding the
parameter regimes that are most suitable for the realization of either static
or flexible Rydberg aggregates based on Li or Rb atoms. The current status of
experimental progress towards enabling Rydberg aggregates is also reviewed.
