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. Close to the critical point, persisting critical anomalies enable the separation of the supercritical state into two different states. 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$_2$, CH$_4$, C$_2$H$_6$, CO$_2$ and H$_2$O at the FL detected by a number of techniques including X-ray, neutron and Raman scattering experiments. %Except for H$_2$O, these experiments were stimulated by the FL and followed the state points of the FL mapped in preceding calculations. We subsequently summarise other developments in the field, including recent extensions of analysis of dynamics at the FL, quantum simulations, topological and geometrical approaches as well as universality of properties at the FL. Finally, we review current theoretical understanding of the supercritical state and list open problems in the field.
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