The experimental study of the dynamics of neutral electron–positron beams is an emerging area of research, enabled by the recent results on the generation of this exotic state of matter in the laboratory. Electron–positron beams and plasmas are believed to play a major role in the dynamics of extreme astrophysical objects such as supermassive black holes and pulsars. For instance, they are believed to be the main constituents of a large number of astrophysical jets, and they have been proposed to significantly contribute to the emission of gamma-ray bursts and their afterglow. However, despite extensive numerical modelling and indirect astrophysical observations, a detailed experimental characterisation of the dynamics of these objects is still at its infancy. Here, we will report on some of the general features of experiments studying the dynamics of electron–positron beams in a fully laser-driven setup.