  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.
