A large and mostly unexplored part of the phase diagram lies above the critical point. The supercritical matter was traditionally believed to be physically homogeneous with no discernible differences between liquidlike and gaslike states. More recently, several proposals have been put forward challenging this view, and here we review the history of this research. About a decade ago, it was proposed that the Frenkel line (FL), corresponding to the dynamical transition of particle motion and related thermodynamic and structural transitions, gives a unique and path-independent way to separate the supercritical states into two qualitatively different states and extends to arbitrarily high pressure and temperature on the phase diagram. Here, we review several lines of enquiry that followed. We focus on the experimental evidence of transitions in deeply supercritical Ne, N, CH4, C2H6, CO2 and H2O at the FL detected by a number of techniques including X-ray, neutron and Raman scattering experiments. We subsequently summarise other developments in the field: recent extensions of analysis of dynamics at the FL, quantum simulations, topological and geometrical approaches, the universality of properties at the FL including transport properties, their fundamental bounds and the implications of the supercritical crossover for astrophysics and planetary science. Finally, we review current theoretical understanding of the supercritical state including its thermodynamic theory and list open problems in the field.