This paper introduces the next generation of fast ignition research facilities now under construction and describes in detail the Japanese project Fast Ignition Realization Experiment (FIREX-I) and its proposed follow-up, FIREX-II. Both the facilities and their scientific objectives are presented. FIREX-I and the other two facilities described in subsequent papers - OMEGA EP at the University of Rochester and the Z-Petawatt at Sandia National Laboratories - will conduct proof-of-principle experiments for the fast ignitor concept. The facilities consist of two components: a long-pulse ( > ns) driver capable of compressing and assembling the fusion fuel and a separate petawatt-class laser for heating. For the FIREX project, the present status of the construction of the 10-kJ-level, high-energy petawatt Laser for Fusion Experiment is reported, and the theoretical basis for high-density plasma heating with an ~10-kJ, 10-ps petawatt laser is discussed to show how this heating pulse is predicted to achieve the plasma parameters required for the fast ignition. The required petawatt spot size, the tolerable carbon fraction in the proposed D-T-loaded foam cryogenic target, appropriate heating laser pulse shape, and the required electron stopping range are explored. The theoretical analysis includes the use of Fokker-Planck simulation to describe the heating of the dense plasma by relativistic electrons created in the petawatt laser-plasma interactions. This modeling indicates that if 30% of the 10-kJ petawatt laser energy is coupled by relativistic electrons into D-T plasmas compressed to 100 to 200 g/cm3, the plasmas will be subsequently heated to 5 keV and fusion gains, defined as fusion energy produced divided by the total incident (compression and heating) laser energy, as high as 0.1 can result.