  With the computational power and algorithmic improvements available today,
the ongoing STAR/RHIC and HADES/GSI experiments, the future FAIR and NICA
facilities becoming operational, and the new precise measurements from NICER
and LIGO/VIRGO, the high-energy nuclear physics and astrophysics communities
are in the unique position to set very stringent constraints on the equation of
state of strongly interacting matter. We review the state-of-the-art of
different approaches used in the description of hot and ultradense baryonic
matter in and out of equilibrium, and discuss the regions in the phase diagram
where heavy-ion collisions and neutron star mergers can overlap. Future
perspectives are discussed to help define a comprehensive, multi-disciplinary
strategy to map out the phase diagram of strongly interacting matter from heavy
ion collisions to neutron stars.
