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 tailored 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 Bose-Einstein condensate 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. © 2018 IOP Publishing Ltd.