  Interactions between quasiparticles mediated by a surrounding environment are
ubiquitous and lead to a range of important effects from collective modes of
low temperature quantum gases, superconductivity, to the interaction between
elementary particles at high energies. This perspective article is motivated by
experimental progress in the fields of quantum degenerate atomic gases, cavity
QED, and two-dimensional (2D) semi-conductors, which enable a systematic
exploration of mediated interactions in new settings and regimes. We first
describe how to microscopically calculate the quasiparticle interaction using
perturbation theory, diagrammatics, and the path integral, highlighting the key
role played by the quantum statistics of the quasiparticles. Recent theoretical
and experimental insights into quasiparticle and mediated interactions in
general obtained from atomic gases are then discussed, after which we focus on
hybrid light-atom systems where a remarkable long range photon mediated
interaction can be realised. Next, we describe new and puzzling results
regarding the interaction between quasiparticles in 2D semiconductors. We then
discuss how mediated interactions open up ways to realise new quantum phases in
atomic and hybrid atom-photon systems as well as 2D semiconductors, and the
perspective ends by posing some open questions and outlook.
