This paper presents a review of the theoretical work that has resulted in a scientific proposal on studies of HighEnergy-Density (HED) states in matter using intense beams of energetic heavy ions that will be available at the future Facility for Antiprotons and Ion Research (FAIR) at Darmstadt [W.F. Henning, Nucl. Inst. Meth B 24 (2003) [725][726][727][728][729]. The proposal is named HEDgeHOB that stands for High Energy Density Matter Generated by Heavy Ion Beams. Two experimental schemes have been worked out for the HEDgeHOB experimental proposal, namely, HIHEX and LAPLAS. The first scheme allows for studies of HED states by isochoric and uniform heating of matter by an intense heavy ion beam that is followed by isentropic expansion of the heated material. Numerical simulations have shown that using the beam parameters that will be available at the FAIR, one can access all the interesting physical states of HED matter including an expanded hot liquid state, twophase liquid-gas region, critical point parameters and strongly coupled plasmas for all the materials of interest. The second scheme involves a low-entropy compression of a test material like hydrogen that is enclosed in a cylindrical shell of a high-Z material like gold or lead. The target can be driven by a hollow or a circular beam. This compression scheme relies on multiple shock reflection between the hydrogen-gold (lead) boundary and the cylinder axis. The hydrodynamic stability of the LAPLAS target has also been analyzed that shows that the implosion is completely stable to Rayleigh-Taylor and Richtmyer-Meshkov instabilities. LAPLAS implosion using a hollow beam is suitable for studying the problem of hydrogen metallization whereas the one employing a circular focal spot leads to physical conditions that are expected to exist in the interiors of the giant planets.