M Middleton, G Lipunova, K Ohsuga, M Abramowicz
Super-critical accretion is of both theoretical and observational importance. Theoretically first established by Shakura & Sunyaev in 1973, studies of this accretion regime has since evolved to encompass vital aspects such as advection, surface effects (neutron stars) and the interplay between the plasma, magnetic fields and radiation in full numerical GR-(R)MHD. In this short review of the subject, we present the underlying analytical theory, observational predictions and the role of numerical simulations, with a focus on super-Eddington accretion onto stellar mass black holes and neutron stars. We then discuss observational progress and tensions, and suggestions for how the field might progress in the coming years.